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            Lecture 16: Counting Rules I
         
    
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<p><strong>Description:</strong> Introduces and defines relationships between sets and covers how they are used to reason about counting.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>
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<p><span m="390">The</span> <span m="520">following</span> <span m="950">content</span> <span m="1550">is</span> <span m="1670">provided</span> <span m="2110">under</span> <span m="2390">a</span> <span m="2430">Creative</span> <span m="2830">Commons</span> <span m="3230">license.</span> <span m="4350">Your</span> <span m="4530">support</span> <span m="5030">will</span> <span m="5200">help</span> <span m="5440">MIT</span> <span m="5890">OpenCourseWare</span> <span m="6680">continue</span> <span m="7190">to</span> <span m="7270">offer</span> <span m="7690">high</span> <span m="7920">quality</span> <span m="8440">educational</span> <span m="9070">resources</span> <span m="9700">for</span> <span m="9850">free.</span> <span m="11050">To</span> <span m="11060">make</span> <span m="11260">a</span> <span m="11310">donation</span> <span m="12000">or</span> <span m="12260">view</span> <span m="12700">additional</span> <span m="13120">materials</span> <span m="13670">from</span> <span m="13810">hundreds</span> <span m="14250">of</span> <span m="14360">MIT</span> <span m="14780">courses,</span> <span m="15900">visit</span> <span m="16120">MIT</span> <span m="16530">OpenCourseWare</span> <span m="17580">at</span> <span m="17750">ocw.mit.edu.</span> </p>
<p><span m="22800">PROFESSOR: So</span> <span m="23040">this</span> <span m="23230">week</span> <span m="23440">we are</span> <span m="23600">going</span> <span m="23810">to</span> <span m="23930">talk</span> <span m="24210">about</span> <span m="24960">counting.</span> <span m="27050">Tonight</span> <span m="27520">is</span> <span m="28410">a</span> <span m="28500">problem</span> <span m="28800">set</span> <span m="29150">eight</span> <span m="29770">due.</span> <span m="32009">For</span> <span m="32200">this</span> <span m="32409">week,</span> <span m="33480">we</span> <span m="33620">will</span> <span m="33760">post</span> <span m="34080">a</span> <span m="34120">new</span> <span m="34260">problem</span> <span m="34670">set</span> <span m="34860">tonight</span> <span m="34910">as</span> <span m="35260">well.</span> <span m="36940">Counting</span> <span m="37420">is</span> <span m="38290">very</span> <span m="38630">important.</span> <span m="39190">The</span> <span m="39300">rest</span> <span m="39510">of</span> <span m="39610">the</span> <span m="39690">semester</span> <span m="40170">after</span> <span m="40460">this</span> <span m="40650">week,</span> <span m="41130">we</span> <span m="41610">will</span> <span m="41780">actually</span> <span m="42150">explain</span> <span m="42960">probability</span> <span m="43560">theory.</span> <span m="44610">And</span> <span m="44770">it's</span> <span m="44950">all</span> <span m="45090">based</span> <span m="45410">on</span> <span m="45520">counting.</span> </p>
<p><span m="48440">So</span> <span m="50460">I'm</span> <span m="50580">going</span> <span m="50760">to</span> <span m="50890">teach</span> <span m="51210">you</span> <span m="51270">this</span> <span m="51470">week</span> <span m="51710">a</span> <span m="51750">whole</span> <span m="51960">toolkit</span> <span m="52650">of all</span> <span m="53000">kinds</span> <span m="53270">of</span> <span m="53420">ways</span> <span m="53760">how</span> <span m="54000">to</span> <span m="54180">count.</span> <span m="54690">And</span> <span m="54830">as</span> <span m="54900">you</span> <span m="55010">can</span> <span m="55170">see,</span> <span m="55770">we're</span> <span m="55930">going</span> <span m="56390">to</span> <span m="56670">talk</span> <span m="56960">about</span> <span m="57270">a</span> <span m="57320">lot</span> <span m="57480">of</span> <span m="57560">different</span> <span m="57880">kinds</span> <span m="58130">of</span> <span m="58270">rules.</span> <span m="59020">A</span> <span m="59160">mapping</span> <span m="59510">rule</span> <span m="59930">we'll</span> <span m="60070">talk</span> <span m="60370">about,</span> <span m="61560">the</span> <span m="62750">pigeon</span> <span m="63100">hole principle,</span> <span m="63510">another</span> <span m="63810">rule,</span> <span m="64620">the</span> <span m="64700">product</span> <span m="65090">rule</span> <span m="65300">and</span> <span m="65379">the</span> <span m="65450">sum</span> <span m="65705">rule.</span> <span m="66620">And</span> <span m="66740">these</span> <span m="66930">are</span> <span m="66990">all</span> <span m="67090">ways</span> <span m="67570">to</span> <span m="67810">count</span> <span m="68840">one</span> <span m="69090">thing</span> <span m="70100">by</span> <span m="70220">counting</span> <span m="70670">another</span> <span m="70770">thing.</span> </p>
<p><span m="71380">Actually,</span> <span m="71730">what</span> <span m="71930">we</span> <span m="72070">are</span> <span m="72150">going</span> <span m="72330">to</span> <span m="72420">talk</span> <span m="72700">about</span> <span m="73040">is</span> <span m="73190">how</span> <span m="73390">to</span> <span m="74590">count</span> <span m="74950">a</span> <span m="75450">difficult</span> <span m="76710">set</span> <span m="77060">the</span> <span m="77870">objects.</span> <span m="79080">And</span> <span m="79250">we</span> <span m="79380">will</span> <span m="79510">map</span> <span m="79770">those</span> <span m="80030">objects</span> <span m="80610">to</span> <span m="80670">something</span> <span m="81130">that</span> <span m="81300">we can</span> <span m="81610">count</span> <span m="81970">in</span> <span m="82080">a</span> <span m="82120">much</span> <span m="82350">easier</span> <span m="82740">way.</span> <span m="85020">OK.</span> <span m="85430">So</span> <span m="85510">let's</span> <span m="86810">start</span> <span m="87200">with</span> <span m="87400">a</span> <span m="87460">lot</span> <span m="87620">of</span> <span m="87700">definitions</span> <span m="88320">actually.</span> </p>
<p><span m="88780">So</span> <span m="89520">we</span> <span m="89640">have</span> <span m="89900">to</span> <span m="90530">talk</span> <span m="90770">about</span> <span m="91080">sets,</span> <span m="91910">sequences,</span> <span m="92650">and</span> <span m="92780">permutations</span> <span m="93980">to</span> <span m="94080">start</span> <span m="94380">off</span> <span m="94590">with.</span> <span m="95960">So</span> <span m="97252">the</span> <span m="97640">definition</span> <span m="98170">of</span> <span m="98300">a</span> <span m="98350">set</span> <span m="98880">is</span> <span m="99810">as</span> <span m="99960">follows.</span> <span m="102640">A</span> <span m="102710">set</span> <span m="103090">is</span> <span m="103280">actually</span> <span m="103730">an</span> <span m="104080">unordered</span> <span m="110320">collection</span> <span m="111450">of</span> <span m="111650">distinct</span> <span m="112450">elements.</span> </p>
<p><span m="121180">So</span> <span m="121440">as</span> <span m="121550">an</span> <span m="121640">example,</span> <span m="124350">we</span> <span m="124450">may</span> <span m="124720">have,</span> <span m="125800">say, a</span> <span m="126140">set</span> <span m="126540">that</span> <span m="126670">contains</span> <span m="128600">the</span> <span m="128889">elements</span> <span m="129330">a,</span> <span m="129550">b,</span> <span m="129740">and</span> <span m="129870">c.</span> <span m="131120">Well,</span> <span m="131370">if</span> <span m="131490">you</span> <span m="131650">reorder</span> <span m="132160">these</span> <span m="132400">elements,</span> <span m="132655">it</span> <span m="132910">doesn't</span> <span m="133180">really</span> <span m="133370">matter.</span> <span m="133690">It's</span> <span m="133810">still</span> <span m="134000">the</span> <span m="134110">same</span> <span m="134400">set.</span> </p>
<p><span m="134980">We</span> <span m="135100">can</span> <span m="135260">also</span> <span m="135530">write</span> <span m="135770">it</span> <span m="135850">as</span> <span m="136690">the</span> <span m="136790">set</span> <span m="137160">c,</span> <span m="137890">a,</span> <span m="138260">b.</span> <span m="139640">On</span> <span m="139780">the</span> <span m="139870">other</span> <span m="140030">hand</span> <span m="140440">if</span> <span m="140480">you</span> <span m="140630">have</span> <span m="141230">a</span> <span m="141400">collection</span> <span m="142230">in</span> <span m="142470">which</span> <span m="142820">say</span> <span m="143710">the</span> <span m="143820">elements</span> <span m="144300">a</span> <span m="145180">appears</span> <span m="145660">twice,</span> <span m="146650">well,</span> <span m="147820">these</span> <span m="148060">two</span> <span m="148220">are</span> <span m="148310">not</span> <span m="148490">distinct.</span> <span m="148970">So</span> <span m="149080">this</span> <span m="149250">is</span> <span m="149350">not</span> <span m="149580">a</span> <span m="149630">set.</span> <span m="151270">So</span> <span m="151450">this</span> <span m="151650">is</span> <span m="151770">not</span> <span m="151980">a</span> <span m="152040">set.</span> </p>
<p><span m="152500">But</span> <span m="152630">it is</span> <span m="152910">a</span> <span m="152960">collection.</span> <span m="154930">And</span> <span m="155210">you</span> <span m="155300">may</span> <span m="155420">call</span> <span m="155610">this</span> <span m="155750">a</span> <span m="155800">multiset.</span> <span m="156175">But</span> <span m="156550">we'll</span> <span m="156630">not</span> <span m="156800">go</span> <span m="156940">into</span> <span m="157140">that.</span> <span m="158350">So</span> <span m="159000">we will</span> <span m="159270">be</span> <span m="159390">talking</span> <span m="159730">about</span> <span m="160020">sets.</span> <span m="160550">And</span> <span m="160770">your</span> <span m="160900">interested</span> <span m="161640">usually</span> <span m="162050">in the</span> <span m="162150">cardinality</span> <span m="162770">of</span> <span m="162880">a</span> <span m="162920">set.</span> </p>
<p><span m="163930">So</span> <span m="164170">what's</span> <span m="164430">that?</span> <span m="165860">The</span> <span m="166010">cardinality</span> <span m="166770">or</span> <span m="166860">size</span> <span m="170590">is</span> <span m="171570">defined</span> <span m="171980">as</span> <span m="172110">follows.</span> <span m="175000">Cardinality</span> <span m="175810">is</span> <span m="175990">just</span> <span m="176240">a</span> <span m="176670">number</span> <span m="177460">of</span> <span m="177660">elements</span> <span m="184470">that</span> <span m="184860">the</span> <span m="184970">set</span> <span m="185400">S</span> <span m="185670">really</span> <span m="185990">has.</span> <span m="186590">So</span> <span m="186770">it's</span> <span m="186990">a</span> <span m="187030">number</span> <span m="188730">of</span> <span m="189360">elements</span> <span m="192790">in</span> <span m="193180">S.</span> </p>
<p><span m="196020">And</span> <span m="196170">how</span> <span m="196340">do</span> <span m="196440">we</span> <span m="196670">denote this/</span> <span m="198970">This is</span> <span m="199090">denoted</span> <span m="199900">by</span> <span m="201100">two</span> <span m="201280">vertical</span> <span m="201720">bars</span> <span m="202550">around</span> <span m="203570">the</span> <span m="203650">letter</span> <span m="204490">that</span> <span m="204760">represents</span> <span m="205460">the</span> <span m="205550">set.</span> <span m="207740">So</span> <span m="208030">we</span> <span m="208240">denote</span> <span m="208430">this</span> <span m="208730">as</span> <span m="208920">follows.</span> </p>
<p><span m="210520">Now,</span> <span m="211510">when</span> <span m="211710">we</span> <span m="211820">talk</span> <span m="212020">about</span> <span m="212260">sets,</span> <span m="212750">we</span> <span m="212910">may</span> <span m="213100">also</span> <span m="213260">be</span> <span m="213390">interested</span> <span m="214220">in</span> <span m="214960">ordered</span> <span m="215850">collection</span> <span m="216450">of</span> <span m="217150">elements.</span> <span m="217940">And</span> <span m="218110">that's</span> <span m="218290">what</span> <span m="218430">we</span> <span m="218560">call</span> <span m="218800">a</span> <span m="218880">sequence.</span> <span m="220820">So</span> <span m="221050">a</span> <span m="221110">sequence</span> <span m="221610">is</span> <span m="221750">defined</span> <span m="222140">as</span> <span m="222260">follows.</span> </p>
<p><span m="224440">A</span> <span m="224560">sequence</span> <span m="228670">is</span> <span m="230860">an</span> <span m="231170">ordered</span> <span m="235050">collection</span> <span m="238580">of</span> <span m="239660">elements.</span> <span m="240170">And</span> <span m="240320">we</span> <span m="240610">also</span> <span m="240910">call</span> <span m="241120">these</span> <span m="241300">elements</span> <span m="243070">components</span> <span m="243980">or</span> <span m="244250">terms.</span> <span m="252810">And</span> <span m="254500">these</span> <span m="254730">elements</span> <span m="255950">do</span> <span m="256050">not</span> <span m="256329">necessarily</span> <span m="256839">have</span> <span m="257010">to</span> <span m="257100">be</span> <span m="257230">distinct--</span> <span m="257899">so</span> <span m="258990">not</span> <span m="259329">necessarily</span> <span m="264730">distinct.</span> </p>
<p><span m="267360">Now</span> <span m="267510">as</span> <span m="267630">an</span> <span m="267690">example,</span> <span m="268270">so</span> <span m="268450">how</span> <span m="268600">do</span> <span m="268700">we</span> <span m="268870">denote</span> <span m="269080">this?</span> <span m="269400">For</span> <span m="269640">sets,</span> <span m="270080">we</span> <span m="270160">use</span> <span m="270770">this</span> <span m="271420">type</span> <span m="271650">of</span> <span m="271750">notation</span> <span m="274490">like</span> <span m="274800">this</span> <span m="274980">symbol.</span> <span m="276520">For</span> <span m="276840">sequences,</span> <span m="277430">we just</span> <span m="277880">have</span> <span m="278080">a</span> <span m="278770">very</span> <span m="279020">simple</span> <span m="279390">type</span> <span m="279610">of</span> <span m="279750">bracket,</span> <span m="280270">just</span> <span m="280740">a</span> <span m="281080">round</span> <span m="281470">bracket.</span> <span m="282630">So</span> <span m="282780">an</span> <span m="282860">example</span> <span m="283500">could</span> <span m="283680">be,</span> <span m="285280">well,</span> <span m="286360">the</span> <span m="286470">elements</span> <span m="287050">a,</span> <span m="287300">b,</span> <span m="287650">and</span> <span m="287850">c.</span> </p>
<p><span m="289240">The</span> <span m="289400">first</span> <span m="289790">entry</span> <span m="290420">or</span> <span m="290520">the</span> <span m="290600">first</span> <span m="291010">term</span> <span m="291280">of</span> <span m="291390">the</span> <span m="291440">sequence--</span> <span m="292750">depending</span> <span m="293090">on</span> <span m="293220">whether</span> <span m="293390">you</span> <span m="293560">look</span> <span m="293760">at</span> <span m="293840">it</span> <span m="293960">from</span> <span m="294350">the left</span> <span m="294670">or the</span> <span m="294945">right,</span> <span m="295220">it</span> <span m="295340">may</span> <span m="295800">differ--</span> <span m="297040">is</span> <span m="297240">here</span> <span m="297530">a,</span> <span m="297840">b,</span> <span m="298110">and then</span> <span m="298410">c.</span> <span m="301140">Another</span> <span m="301450">one</span> <span m="301720">could</span> <span m="301940">be</span> <span m="302310">a,</span> <span m="303000">b,</span> <span m="303340">and</span> <span m="303550">a.</span> <span m="303810">And</span> <span m="304070">as</span> <span m="304130">you</span> <span m="304260">can</span> <span m="304410">see,</span> <span m="304630">the</span> <span m="304750">element</span> <span m="305120">a</span> <span m="305550">occurs</span> <span m="305900">twice</span> <span m="306290">in</span> <span m="306370">the</span> <span m="306470">sequence.</span> </p>
<p><span m="310700">So</span> <span m="310890">we're</span> <span m="311010">going</span> <span m="311250">to</span> <span m="311470">relate</span> <span m="312020">sets</span> <span m="312375">and</span> <span m="312730">sequences.</span> <span m="313980">And</span> <span m="315720">let's</span> <span m="315940">talk</span> <span m="316170">about</span> <span m="316460">the</span> <span m="316540">permutation.</span> <span m="318080">So</span> <span m="319150">a</span> <span m="319250">permutation</span> <span m="320000">of a set</span> <span m="320540">is</span> <span m="320660">defined</span> <span m="321030">as</span> <span m="321160">follows.</span> <span m="328500">A</span> <span m="328610">permutation</span> <span m="330833">of</span> <span m="332190">a</span> <span m="332250">set</span> <span m="333990">S</span> <span m="335430">is</span> <span m="335730">actually</span> <span m="336080">a</span> <span m="336130">sequence</span> <span m="341430">that</span> <span m="341630">contains</span> <span m="342320">every</span> <span m="342440">element</span> <span m="342880">in</span> <span m="343000">S</span> <span m="343420">exactly</span> <span m="343710">once.</span> </p>
<p><span m="355250">So</span> <span m="355760">every</span> <span m="355850">element</span> <span m="356260">in</span> <span m="356410">S</span> <span m="357060">occurs</span> <span m="357570">exactly</span> <span m="358200">once.</span> <span m="363280">And</span> <span m="364500">as</span> <span m="364740">an</span> <span m="364930">example,</span> <span m="366360">we</span> <span m="366370">may</span> <span m="366580">look</span> <span m="366990">at</span> <span m="367250">the</span> <span m="367350">set</span> <span m="367690">that</span> <span m="367830">we</span> <span m="368030">have</span> <span m="368260">described</span> <span m="368760">over</span> <span m="369000">here, the</span> <span m="369230">set</span> <span m="369600">a,b,</span> <span m="370095">and c.</span> <span m="371930">And</span> <span m="373360">how</span> <span m="373570">many</span> <span m="374520">permutations</span> <span m="375460">are</span> <span m="375720">there?</span> </p>
<p><span m="376530">So</span> <span m="377390">the</span> <span m="377450">first</span> <span m="377810">one</span> <span m="378690">I</span> <span m="378910">may</span> <span m="380560">just</span> <span m="381040">order</span> <span m="381450">the</span> <span m="381580">elements</span> <span m="382140">in</span> <span m="382390">a, b,</span> <span m="382736">and c.</span> <span m="383430">So</span> <span m="383700">I</span> <span m="383830">have,</span> <span m="384030">say, the</span> <span m="384330">sequence</span> <span m="384670">a</span> <span m="385010">and then</span> <span m="385730">b,</span> <span m="386070">and</span> <span m="386230">then</span> <span m="386420">c.</span> <span m="388020">There</span> <span m="388220">are</span> <span m="388430">many</span> <span m="388720">more.</span> <span m="389330">I</span> <span m="389510">can,</span> <span m="390280">for</span> <span m="390370">example,</span> <span m="390730">start</span> <span m="391100">with</span> <span m="391320">b,</span> <span m="393040">and</span> <span m="393280">then</span> <span m="393480">c,</span> <span m="393990">and</span> <span m="394140">then</span> <span m="394160">may</span> <span m="394350">cycle</span> <span m="394760">through</span> <span m="395020">to</span> <span m="395320">a.</span> <span m="396530">That's</span> <span m="396720">another</span> <span m="397700">permutation</span> <span m="398450">of</span> <span m="398670">these</span> <span m="398950">three</span> <span m="399200">elements.</span> </p>
<p><span m="400880">And</span> <span m="401050">I</span> <span m="401090">can</span> <span m="401250">do</span> <span m="401330">this</span> <span m="401490">once</span> <span m="401750">more.</span> <span m="402010">I can,</span> <span m="402280">for</span> <span m="402470">example,</span> <span m="403760">start</span> <span m="404260">with</span> <span m="405040">c,</span> <span m="406340">and</span> <span m="406510">then</span> <span m="406780">a,</span> <span m="407115">and</span> <span m="407450">then</span> <span m="407640">b.</span> <span m="409760">And</span> <span m="409890">it</span> <span m="409970">turns</span> <span m="410290">out</span> <span m="410450">that</span> <span m="410550">we</span> <span m="410650">can</span> <span m="410800">do</span> <span m="410930">a</span> <span m="411000">few</span> <span m="411220">more.</span> <span m="413150">You</span> <span m="413240">can</span> <span m="413390">also</span> <span m="413600">start</span> <span m="413970">to</span> <span m="414080">c,</span> <span m="414630">and</span> <span m="414810">then</span> <span m="415020">b,</span> <span m="415410">and</span> <span m="415580">then</span> <span m="415810">a.</span> <span m="416140">We sort</span> <span m="416390">of</span> <span m="418630">reversed</span> <span m="418960">the</span> <span m="419360">order</span> <span m="419810">that</span> <span m="419930">we</span> <span m="420140">had</span> <span m="420290">over</span> <span m="420570">here</span> <span m="420960">into</span> <span m="421620">its</span> <span m="421810">opposite,</span> <span m="423100">so</span> <span m="423270">first</span> <span m="423520">c,</span> <span m="423780">then</span> <span m="423950">b,</span> <span m="424100">then</span> <span m="424360">a.</span> </p>
<p><span m="425070">And</span> <span m="425200">we</span> <span m="425270">can</span> <span m="425400">look</span> <span m="425570">at</span> <span m="425630">this</span> <span m="425790">cycle</span> <span m="426850">there.</span> <span m="427390">There's</span> <span m="427600">shifts.</span> <span m="428480">And</span> <span m="428910">we</span> <span m="429000">start</span> <span m="429500">with</span> <span m="429610">b,</span> <span m="429890">a,</span> <span m="430330">and then we</span> <span m="430530">rotate</span> <span m="430950">through</span> <span m="431260">to</span> <span m="431410">c,</span> <span m="432730">which</span> <span m="432900">is</span> <span m="432970">what</span> <span m="433090">we</span> <span m="433310">did</span> <span m="433550">over</span> <span m="433840">here</span> <span m="434060">when</span> <span m="434170">you</span> <span m="434320">went</span> <span m="434500">from</span> <span m="434680">this</span> <span m="435070">permutation</span> <span m="435660">to</span> <span m="435780">this</span> <span m="436010">one.</span> </p>
<p><span m="437050">So</span> <span m="437100">we</span> <span m="437190">have</span> <span m="437350">to</span> <span m="437420">b,</span> <span m="438040">a,</span> <span m="438370">and then</span> <span m="438725">c.</span> <span m="439700">And</span> <span m="439870">then</span> <span m="440000">we</span> <span m="440110">may</span> <span m="440770">start</span> <span m="441280">with</span> <span m="441520">a,</span> <span m="442432">and</span> <span m="442890">we</span> <span m="443010">have</span> <span m="443300">a,</span> <span m="443683">c,</span> <span m="444066">and</span> <span m="444450">b.</span> <span m="445410">It</span> <span m="445510">turns</span> <span m="445880">out</span> <span m="446200">that</span> <span m="446600">this</span> <span m="446820">is</span> <span m="447090">actually</span> <span m="448910">all</span> <span m="449120">the</span> <span m="449210">possible</span> <span m="449610">permutations</span> <span m="451100">of</span> <span m="451880">this set.</span> <span m="452100">So</span> <span m="452310">there's</span> <span m="452510">six</span> <span m="453650">of</span> <span m="453700">these.</span> </p>
<p><span m="456202">And</span> <span m="456700">in</span> <span m="456860">general,</span> <span m="458850">how</span> <span m="458980">many</span> <span m="459330">permutations</span> <span m="459950">are</span> <span m="460200">there</span> <span m="460660">of a set</span> <span m="461160">of</span> <span m="461330">n</span> <span m="461560">elements?</span> <span m="462760">So</span> <span m="462860">let's</span> <span m="463060">have</span> <span m="463250">a</span> <span m="463310">look</span> <span m="463560">here</span> <span m="464450">what</span> <span m="464650">we</span> <span m="464780">did.</span> <span m="466340">So</span> <span m="467070">if</span> <span m="467360">I</span> <span m="467480">want</span> <span m="467690">to</span> <span m="467900">create</span> <span m="468140">a</span> <span m="468200">permutation</span> <span m="468940">of</span> <span m="469060">a</span> <span m="469130">set,</span> <span m="470020">I</span> <span m="470110">may</span> <span m="470290">start</span> <span m="470680">off</span> <span m="471070">by</span> <span m="471740">selecting</span> <span m="472280">for</span> <span m="472430">the</span> <span m="472530">first</span> <span m="473000">term</span> <span m="474130">a,</span> <span m="475170">and</span> <span m="475250">b,</span> <span m="475580">or</span> <span m="475810">c.</span> </p>
<p><span m="476110">So</span> <span m="476250">I have</span> <span m="476430">actually</span> <span m="476800">three</span> <span m="477100">choices.</span> <span m="478800">So</span> <span m="478960">the</span> <span m="479070">first</span> <span m="479540">term</span> <span m="480390">has</span> <span m="485380">three</span> <span m="485670">choices.</span> <span m="489200">The</span> <span m="489270">second</span> <span m="489710">term</span> <span m="490010">over</span> <span m="490300">here,</span> <span m="490620">well,</span> <span m="491560">for</span> <span m="491710">example,</span> <span m="492010">suppose</span> <span m="492370">my</span> <span m="492480">first</span> <span m="492900">term</span> <span m="493200">was</span> <span m="493550">b.</span> <span m="494910">Then</span> <span m="495580">the</span> <span m="495720">second</span> <span m="496180">term,</span> <span m="497300">well,</span> <span m="497510">must</span> <span m="497780">be</span> <span m="497940">an</span> <span m="498090">element</span> <span m="498480">that</span> <span m="498570">is</span> <span m="498740">different</span> <span m="499090">from</span> <span m="499260">b,</span> <span m="499550">as</span> <span m="499840">yet</span> <span m="500140">part</span> <span m="500380">of</span> <span m="500690">S.</span> </p>
<p><span m="501880">So</span> <span m="502500">I</span> <span m="502640">have</span> <span m="502840">only</span> <span m="503150">two</span> <span m="503370">choices</span> <span m="503890">for</span> <span m="504540">the</span> <span m="504620">second</span> <span m="504980">term,</span> <span m="505200">either</span> <span m="505450">c</span> <span m="506070">or</span> <span m="506350">a.</span> <span m="507450">So</span> <span m="507600">the</span> <span m="507710">second</span> <span m="509780">term</span> <span m="510140">has</span> <span m="510520">two</span> <span m="510770">choices.</span> <span m="512400">And</span> <span m="513179">once</span> <span m="513500">I have</span> <span m="513700">chosen</span> <span m="514100">the</span> <span m="514179">second</span> <span m="514570">one,</span> <span m="514990">well,</span> <span m="515309">if</span> <span m="515549">I've</span> <span m="515850">already</span> <span m="516289">chosen</span> <span m="516750">b and</span> <span m="517039">c,</span> <span m="517770">there's</span> <span m="517960">only</span> <span m="518230">one</span> <span m="518460">element</span> <span m="518809">left</span> <span m="519100">in</span> <span m="519179">the</span> <span m="519260">set</span> <span m="520390">over</span> <span m="520700">here,</span> <span m="521400">only</span> <span m="521760">a.</span> <span m="522150">So I have</span> <span m="522250">only</span> <span m="522520">one</span> <span m="522799">choice</span> <span m="523200">left.</span> </p>
<p><span m="524480">So</span> <span m="525470">in</span> <span m="525640">this</span> <span m="525900">way,</span> <span m="526080">we</span> <span m="526210">may</span> <span m="526450">count</span> <span m="526860">the</span> <span m="527010">total</span> <span m="527370">number</span> <span m="527740">of</span> <span m="527880">permutations</span> <span m="528740">of</span> <span m="528880">this</span> <span m="529090">set</span> <span m="529390">as</span> <span m="529970">the</span> <span m="530060">number</span> <span m="530280">of</span> <span m="530410">choices</span> <span m="530820">that</span> <span m="530990">I</span> <span m="531110">have</span> <span m="531320">for</span> <span m="531450">the</span> <span m="531530">first</span> <span m="532180">term</span> <span m="532720">times</span> <span m="533220">the</span> <span m="533300">number</span> <span m="533830">of</span> <span m="533950">choices</span> <span m="534270">for</span> <span m="534380">the</span> <span m="534460">second</span> <span m="534850">term,</span> <span m="535430">3</span> <span m="535690">times</span> <span m="536070">2</span> <span m="536980">times</span> <span m="537350">the</span> <span m="537430">number</span> <span m="537670">of</span> <span m="537790">choices</span> <span m="538230">for</span> <span m="538340">the</span> <span m="538430">third</span> <span m="538810">term,</span> <span m="539100">which</span> <span m="539230">is</span> <span m="539340">only</span> <span m="539640">one.</span> <span m="540520">So</span> <span m="540710">the</span> <span m="540840">third</span> <span m="543200">term</span> <span m="548790">has</span> <span m="549020">only</span> <span m="549310">one</span> <span m="549510">choice.</span> </p>
<p><span m="550540">Now</span> <span m="550700">in</span> <span m="550800">general,</span> <span m="552410">we</span> <span m="552550">can</span> <span m="552720">do</span> <span m="552860">this</span> <span m="553050">for</span> <span m="553270">any</span> <span m="553640">permutation.</span> <span m="554520">And</span> <span m="556380">if</span> <span m="556580">you</span> <span m="556720">want</span> <span m="556890">to</span> <span m="557010">count</span> <span m="557350">the</span> <span m="557430">number</span> <span m="557700">of</span> <span m="557900">permutations,</span> <span m="561860">of</span> <span m="563730">a</span> <span m="563880">set</span> <span m="565400">with</span> <span m="567220">n</span> <span m="568000">elements,</span> <span m="570670">well,</span> <span m="570900">it</span> <span m="571020">turns</span> <span m="571330">out</span> <span m="571470">that</span> <span m="571650">it's</span> <span m="571800">equal</span> <span m="572140">to</span> <span m="572460">n</span> <span m="572790">times</span> <span m="573160">n</span> <span m="573360">minus</span> <span m="573770">1</span> <span m="573960">just</span> <span m="574210">like</span> <span m="574430">here.</span> <span m="574920">We</span> <span m="575090">have</span> <span m="575470">three</span> <span m="575770">elements,</span> <span m="576270">3</span> <span m="576520">times</span> <span m="577190">3</span> <span m="577470">minus</span> <span m="577840">1</span> <span m="578120">and</span> <span m="578330">so</span> <span m="578530">on,</span> <span m="578990">n</span> <span m="579270">times</span> <span m="579550">n</span> <span m="579710">minus</span> <span m="580050">1</span> <span m="580230">times</span> <span m="580500">n</span> <span m="580660">minus</span> <span m="581030">2,</span> <span m="581380">et</span> <span m="581540">cetera</span> <span m="585980">all</span> <span m="586090">the</span> <span m="586180">way</span> <span m="586590">up</span> <span m="586750">to</span> <span m="586940">one.</span> </p>
<p><span m="587620">And</span> <span m="587880">this</span> <span m="588090">is</span> <span m="588310">n</span> <span m="588510">factorial.</span> <span m="589550">And</span> <span m="589650">you've</span> <span m="589780">seen</span> <span m="590020">this</span> <span m="590220">already</span> <span m="591110">when</span> <span m="591290">we</span> <span m="591430">talked</span> <span m="591730">about</span> <span m="591980">Stirling's</span> <span m="592380">formula</span> <span m="593180">and</span> <span m="593310">how to</span> <span m="593530">approximate</span> <span m="593866">this.</span> <span m="595731">Now,</span> <span m="596130">this</span> <span m="596350">type</span> <span m="596600">of</span> <span m="596750">reasoning</span> <span m="597370">we will</span> <span m="597620">generalize</span> <span m="598260">later</span> <span m="598610">on</span> <span m="599070">when</span> <span m="599230">we</span> <span m="599340">come</span> <span m="599570">to</span> <span m="599730">the</span> <span m="599850">Generalized</span> <span m="601110">Product</span> <span m="601600">rule.</span> <span m="602770">And</span> <span m="603980">But</span> <span m="604100">this</span> <span m="604270">is</span> <span m="604410">already</span> <span m="604700">a</span> <span m="604760">first</span> <span m="605070">example</span> <span m="605590">of</span> <span m="605670">how</span> <span m="605880">we</span> <span m="605980">go</span> <span m="606160">about</span> <span m="606500">this.</span> </p>
<p><span m="608350">So</span> <span m="608450">permutations</span> <span m="609120">relates</span> <span m="610500">sets</span> <span m="610880">and</span> <span m="611050">sequences.</span> <span m="613470">So</span> <span m="613520">now</span> <span m="613700">we</span> <span m="613880">go</span> <span m="614080">on</span> <span m="614350">to</span> <span m="614730">define</span> <span m="615200">more</span> <span m="615380">special</span> <span m="616120">functions.</span> <span m="616970">So</span> <span m="617180">permutations</span> <span m="617590">is</span> <span m="618000">one</span> <span m="618250">kind</span> <span m="618490">of</span> <span m="618580">mapping.</span> </p>
<p><span m="619640">So</span> <span m="619760">let's</span> <span m="620000">now</span> <span m="620160">define</span> <span m="621050">functions.</span> <span m="622770">And</span> <span m="623910">then</span> <span m="624070">we will</span> <span m="624600">talk</span> <span m="624860">about</span> <span m="625210">a</span> <span m="625260">few</span> <span m="625480">different</span> <span m="626350">flavors</span> <span m="626910">of</span> <span m="627050">those.</span> <span m="629440">We</span> <span m="629620">talk</span> <span m="629870">about</span> <span m="630100">surjective</span> <span m="630640">functions,</span> <span m="631110">injective</span> <span m="631530">functions,</span> <span m="632070">and</span> <span m="632270">bijective</span> <span m="632630">functions.</span> </p>
<p><span m="633920">And</span> <span m="634460">the</span> <span m="634650">whole</span> <span m="634870">idea</span> <span m="635165">is</span> <span m="635460">is</span> <span m="635730">that if</span> <span m="636000">I</span> <span m="636710">can</span> <span m="637080">use</span> <span m="638370">a</span> <span m="638420">mapping</span> <span m="638880">from</span> <span m="639100">one</span> <span m="639360">set</span> <span m="639700">to</span> <span m="639940">another</span> <span m="640230">set</span> <span m="641960">that</span> <span m="642650">satisfies</span> <span m="643260">some</span> <span m="643500">of</span> <span m="643600">those</span> <span m="643850">properties,</span> <span m="644115">it</span> <span m="644380">can</span> <span m="644530">say</span> <span m="644810">something</span> <span m="645600">about</span> <span m="645950">how</span> <span m="646160">their</span> <span m="646590">cardinalities</span> <span m="647330">are</span> <span m="647440">related.</span> <span m="648630">And</span> <span m="648770">that's</span> <span m="648920">what</span> <span m="649060">we</span> <span m="649200">want.</span> <span m="649680">We want to</span> <span m="649780">count.</span> </p>
<p><span m="651440">OK</span> <span m="651710">So</span> <span m="651890">the</span> <span m="652010">definition</span> <span m="653930">of a</span> <span m="654060">function</span> <span m="655300">is</span> <span m="655550">as</span> <span m="655740">follows.</span> <span m="660220">So</span> <span m="660350">a</span> <span m="660420">function</span> <span m="661870">f</span> <span m="662910">from</span> <span m="663250">x</span> <span m="663670">to</span> <span m="663900">y</span> <span m="665050">is</span> <span m="665320">actually</span> <span m="666050">a</span> <span m="666760">relation</span> <span m="670750">between</span> <span m="671700">the</span> <span m="671810">sets</span> <span m="672350">X</span> <span m="672710">and</span> <span m="672940">Y.</span> <span m="678730">And</span> <span m="679160">we</span> <span m="679250">say</span> <span m="679590">that--</span> <span m="681520">oh--</span> <span m="681990">with</span> <span m="682180">the</span> <span m="682360">property</span> <span m="684190">that</span> <span m="684850">every</span> <span m="685480">single</span> <span m="685830">element,</span> <span m="687090">every</span> <span m="687520">element,</span> <span m="690530">of</span> <span m="690830">X</span> <span m="691890">is</span> <span m="692130">actually</span> <span m="692560">related</span> <span m="693020">to</span> <span m="693260">exactly</span> <span m="693610">one</span> <span m="693860">element</span> <span m="694680">of</span> <span m="694820">Y.</span> </p>
<p><span m="710070">And</span> <span m="712310">we</span> <span m="712470">will</span> <span m="712670">call</span> <span m="713490">x</span> <span m="713860">to</span> <span m="714010">be</span> <span m="714300">the</span> <span m="715200">domain</span> <span m="718490">of</span> <span m="718680">the</span> <span m="718750">function</span> <span m="719170">f.</span> <span m="719920">And</span> <span m="720230">Y</span> <span m="720490">we</span> <span m="720650">will</span> <span m="720810">call</span> <span m="721100">the</span> <span m="721210">range</span> <span m="722780">or</span> <span m="722970">image</span> <span m="723680">of</span> <span m="724280">the</span> <span m="724380">function</span> <span m="724740">f.</span> <span m="726860">So</span> <span m="727060">let's</span> <span m="727290">give</span> <span m="727440">an</span> <span m="727530">example.</span> <span m="728670">And</span> <span m="729020">to</span> <span m="729170">see</span> <span m="730550">a</span> <span m="730650">couple</span> <span m="730900">of</span> <span m="730990">examples</span> <span m="732180">of,</span> <span m="732830">first</span> <span m="733070">of all,</span> <span m="733400">a</span> <span m="733530">function</span> <span m="734240">and</span> <span m="734470">then</span> <span m="736020">relations</span> <span m="736600">to</span> <span m="736690">that are</span> <span m="736840">actually</span> <span m="737190">not</span> <span m="737410">functions.</span> </p>
<p><span m="738680">So</span> <span m="739550">suppose</span> <span m="739940">you</span> <span m="740040">have</span> <span m="740250">a</span> <span m="740300">mapping</span> <span m="740900">from</span> <span m="741210">x</span> <span m="741570">which</span> <span m="741820">just</span> <span m="742080">contains</span> <span m="742980">the</span> <span m="743160">elements</span> <span m="743610">a,</span> <span m="743820">b,</span> <span m="744020">and</span> <span m="744170">c,</span> <span m="744500">just</span> <span m="744790">in line</span> <span m="745130">with this</span> <span m="745360">example</span> <span m="745880">over</span> <span m="746110">here.</span> <span m="746795">And</span> <span m="747070">we</span> <span m="747160">have</span> <span m="747330">a</span> <span m="747390">mapping</span> <span m="747860">f</span> <span m="748920">that</span> <span m="749270">maps</span> <span m="749760">to</span> <span m="753620">the</span> <span m="753820">set</span> <span m="754080">y.</span> <span m="754410">And</span> <span m="754540">the</span> <span m="754640">y</span> <span m="754840">is</span> <span m="755030">just</span> <span m="755320">the</span> <span m="755400">numbers</span> <span m="755790">1,</span> <span m="756070">2,</span> <span m="756380">and</span> <span m="756580">3.</span> </p>
<p><span m="757840">Well,</span> <span m="758040">i could map,</span> <span m="758500">for</span> <span m="758660">example,</span> <span m="759210">a</span> <span m="759750">to</span> <span m="760470">1,</span> <span m="761720">b</span> <span m="762450">to</span> <span m="762980">3,</span> <span m="763860">and</span> <span m="764090">c</span> <span m="764540">to</span> <span m="765000">3.</span> <span m="766050">Now</span> <span m="766490">this</span> <span m="766680">is</span> <span m="766840">a</span> <span m="767140">function.</span> <span m="767660">Because</span> <span m="769050">every</span> <span m="769470">element</span> <span m="769910">of</span> <span m="770060">x</span> <span m="770640">is</span> <span m="770810">met</span> <span m="771070">to</span> <span m="771330">exactly</span> <span m="771720">one</span> <span m="772050">element</span> <span m="772450">of</span> <span m="772570">y.</span> <span m="772880">a</span> <span m="773230">is</span> <span m="773650">just</span> <span m="773950">mapped</span> <span m="774240">to</span> <span m="774420">1.</span> <span m="775360">b</span> <span m="775640">is</span> <span m="775780">also</span> <span m="776040">mapped</span> <span m="776240">to</span> <span m="776320">an</span> <span m="776390">element,</span> <span m="776940">only</span> <span m="777230">3.</span> <span m="778120">c is</span> <span m="778530">mapped</span> <span m="779060">to</span> <span m="779470">3</span> <span m="779750">as</span> <span m="779890">well.</span> </p>
<p><span m="780930">And they</span> <span m="781110">usually</span> <span m="781600">write</span> <span m="781880">this</span> <span m="782310">as</span> <span m="783430">f</span> <span m="785160">evaluated</span> <span m="785470">in</span> <span m="785770">a</span> <span m="786180">is</span> <span m="786340">equal</span> <span m="786640">to</span> <span m="786770">1.</span> <span m="787730">And</span> <span m="787950">f</span> <span m="788250">b</span> <span m="789350">is</span> <span m="789540">equal</span> <span m="789780">to</span> <span m="789910">3.</span> <span m="791090">And</span> <span m="791340">f</span> <span m="791590">of</span> <span m="791960">c</span> <span m="792380">is</span> <span m="793110">equal</span> <span m="793370">to</span> <span m="793480">3</span> <span m="793730">as</span> <span m="793840">well.</span> </p>
<p><span m="794960">Now</span> <span m="794990">what</span> <span m="795130">is</span> <span m="795250">not</span> <span m="795420">a</span> <span m="795480">function--</span> <span m="796190">oh,</span> <span m="796590">I could, for</span> <span m="796750">example,</span> <span m="797210">add</span> <span m="797380">another</span> <span m="797920">edge</span> <span m="798230">over</span> <span m="798480">here</span> <span m="798710">if</span> <span m="798860">I</span> <span m="798890">wanted</span> <span m="799290">to.</span> <span m="800440">But</span> <span m="800470">this</span> <span m="800620">is</span> <span m="800750">not</span> <span m="800900">a</span> <span m="800950">function.</span> <span m="801330">Because</span> <span m="802100">b is</span> <span m="802410">now</span> <span m="802840">mapped to</span> <span m="803020">two</span> <span m="803270">elements,</span> <span m="804030">2</span> <span m="804320">and</span> <span m="804600">3.</span> <span m="805440">And</span> <span m="805600">that's</span> <span m="805810">not</span> <span m="806090">what's</span> <span m="806400">covered</span> <span m="806710">by</span> <span m="806850">this</span> <span m="807060">definition.</span> <span m="807610">So</span> <span m="807760">this</span> <span m="807940">is</span> <span m="808080">not</span> <span m="808250">a</span> <span m="808310">function.</span> </p>
<p><span m="810050">I</span> <span m="810140">can</span> <span m="810330">also</span> <span m="810590">remove,</span> <span m="811570">say,</span> <span m="811950">an</span> <span m="812130">edge.</span> <span m="812670">Well,</span> <span m="812830">in</span> <span m="812920">this</span> <span m="813150">case,</span> <span m="813450">b</span> <span m="813680">is</span> <span m="813840">not</span> <span m="814020">mapped</span> <span m="814220">to</span> <span m="814380">anything</span> <span m="814870">at</span> <span m="814960">all.</span> <span m="815380">And</span> <span m="815420">that's</span> <span m="815600">not</span> <span m="815750">a</span> <span m="815800">function</span> <span m="816190">either.</span> <span m="817050">So</span> <span m="817240">we</span> <span m="817400">really</span> <span m="817770">have</span> <span m="818810">the</span> <span m="818940">property</span> <span m="819420">for</span> <span m="819540">functions</span> <span m="820180">that</span> <span m="820370">there</span> <span m="820510">is</span> <span m="820560">six</span> <span m="820850">exactly</span> <span m="821240">one</span> <span m="821550">outgoing</span> <span m="822120">arrow,</span> <span m="822520">if</span> <span m="822610">you</span> <span m="822720">want</span> <span m="822900">to</span> <span m="823000">think</span> <span m="823230">about</span> <span m="823480">it</span> <span m="823590">as</span> <span m="823720">being a</span> <span m="824020">graph,</span> <span m="827740">from</span> <span m="827960">each</span> <span m="828280">element</span> <span m="828730">in</span> <span m="828880">x</span> <span m="829360">to</span> <span m="829510">exactly</span> <span m="830500">one</span> <span m="830850">element</span> <span m="831450">in</span> <span m="832330">y.</span> </p>
<p><span m="834680">So</span> <span m="834880">now</span> <span m="835060">we</span> <span m="835180">can</span> <span m="835600">talk</span> <span m="835920">about</span> <span m="836660">a</span> <span m="836740">few</span> <span m="836970">definitions.</span> <span m="841580">So</span> <span m="841940">we</span> <span m="842110">will</span> <span m="842280">talk</span> <span m="842920">about</span> <span m="843550">these</span> <span m="843740">few</span> <span m="843990">properties,</span> <span m="844910">surjective,</span> <span m="845400">injective,</span> <span m="846040">and</span> <span m="846430">bijective.</span> <span m="847280">And</span> <span m="847470">then</span> <span m="848180">we</span> <span m="848300">can</span> <span m="848420">start</span> <span m="848890">to do</span> <span m="849120">a few</span> <span m="850030">interesting</span> <span m="851140">examples.</span> </p>
<p><span m="852770">So</span> <span m="853750">a</span> <span m="853830">function</span> <span m="854280">f</span> <span m="855430">that</span> <span m="855600">goes</span> <span m="855850">from</span> <span m="855990">x</span> <span m="856220">to</span> <span m="856390">y</span> <span m="856740">is called</span> <span m="857190">surjective.</span> <span m="861640">if</span> <span m="865020">every</span> <span m="865390">single</span> <span m="865740">element</span> <span m="872390">of</span> <span m="872490">y</span> <span m="874450">is</span> <span m="874870">mapped</span> <span m="879800">to</span> <span m="880070">at</span> <span m="880220">least</span> <span m="884690">once.</span> <span m="885190">So</span> <span m="885380">what</span> <span m="885470">does</span> <span m="885640">that</span> <span m="885740">mean?</span> <span m="887840">To</span> <span m="888220">at</span> <span m="888430">least</span> <span m="888940">once.</span> </p>
<p><span m="893170">So</span> <span m="893570">every</span> <span m="893910">element</span> <span m="894550">of</span> <span m="894770">y,</span> <span m="895650">so</span> <span m="896980">say</span> <span m="897980">1</span> <span m="898290">for</span> <span m="898450">example</span> <span m="898850">here,</span> <span m="899380">is</span> <span m="899580">mapped</span> <span m="899990">to</span> <span m="900230">at</span> <span m="900390">least</span> <span m="900730">once.</span> <span m="901340">Well,</span> <span m="903010">to</span> <span m="903700">the</span> <span m="903880">element</span> <span m="904200">1</span> <span m="904500">we</span> <span m="904690">have</span> <span m="904960">mapped</span> <span m="905460">a.</span> <span m="905790">So</span> <span m="905910">that's</span> <span m="906140">great.</span> <span m="906520">But</span> <span m="906680">for</span> <span m="906770">example</span> <span m="907210">element</span> <span m="907600">2</span> <span m="908390">is</span> <span m="908580">not</span> <span m="908800">mapped</span> <span m="909050">to</span> <span m="909520">at</span> <span m="909660">all.</span> </p>
<p><span m="910710">So</span> <span m="910990">this</span> <span m="911180">particular</span> <span m="911540">example</span> <span m="912010">is</span> <span m="912130">not</span> <span m="912330">surjective.</span> <span m="913070">But</span> <span m="913210">we will</span> <span m="913430">come</span> <span m="913600">to</span> <span m="913720">a</span> <span m="913780">few</span> <span m="913930">examples</span> <span m="914420">that</span> <span m="914560">are.</span> <span m="915240">So</span> <span m="915640">here</span> <span m="916000">we</span> <span m="916110">have</span> <span m="916420">the</span> <span m="916510">distinction</span> <span m="916810">that</span> <span m="917110">every</span> <span m="917800">element</span> <span m="918320">of</span> <span m="918530">y,</span> <span m="918790">so every</span> <span m="918970">single</span> <span m="919410">element</span> <span m="919730">of</span> <span m="919840">y,</span> <span m="920120">is</span> <span m="920270">mapped</span> <span m="920580">to</span> <span m="921730">at</span> <span m="921860">least</span> <span m="922170">once.</span> </p>
<p><span m="923920">The</span> <span m="924040">injective</span> <span m="926270">is</span> <span m="926540">defined</span> <span m="927370">in</span> <span m="927500">a</span> <span m="927550">similar</span> <span m="928020">fashion.</span> <span m="930160">But</span> <span m="930350">now,</span> <span m="930920">every</span> <span m="931200">element</span> <span m="931590">of</span> <span m="931710">y</span> <span m="931930">is</span> <span m="932080">not</span> <span m="932240">mapped</span> <span m="932480">to</span> <span m="933220">at</span> <span m="933390">least</span> <span m="933730">once,</span> <span m="934290">but</span> <span m="934840">at</span> <span m="935040">most</span> <span m="935420">once.</span> <span m="940110">So</span> <span m="940170">let's</span> <span m="940350">have</span> <span m="940500">look</span> <span m="940740">over</span> <span m="940990">here.</span> </p>
<p><span m="941550">And</span> <span m="941850">that's</span> <span m="941960">also</span> <span m="942170">not</span> <span m="942350">true</span> <span m="942510">for</span> <span m="942640">this</span> <span m="942830">example</span> <span m="943350">actually.</span> <span m="943660">Because</span> <span m="944510">three</span> <span m="945710">is</span> <span m="945970">mapped</span> <span m="946340">to</span> <span m="946980">two</span> <span m="947440">times.</span> <span m="948120">So</span> <span m="948320">it's</span> <span m="948440">not</span> <span m="948660">mapped</span> <span m="948910">to</span> <span m="949050">at</span> <span m="949190">most</span> <span m="949560">once.</span> <span m="950250">So</span> <span m="950380">this</span> <span m="950560">example</span> <span m="951010">is</span> <span m="951150">also</span> <span m="951460">not</span> <span m="951730">injective.</span> <span m="953510">Because</span> <span m="954470">if</span> <span m="954630">the</span> <span m="954700">function</span> <span m="954960">is</span> <span m="955070">injective,</span> <span m="955620">every</span> <span m="955880">element</span> <span m="956290">of</span> <span m="957020">y</span> <span m="957350">of</span> <span m="957470">the</span> <span m="957580">range</span> <span m="957960">is</span> <span m="958070">mapped</span> <span m="958330">to</span> <span m="958550">at</span> <span m="958670">most</span> <span m="958970">once.</span> </p>
<p><span m="960350">Bijective</span> <span m="962090">is</span> <span m="964880">if</span> <span m="966770">every</span> <span m="966990">element</span> <span m="967500">of</span> <span m="967640">y</span> <span m="967920">is</span> <span m="968110">mapped</span> <span m="968370">to</span> <span m="968610">exactly</span> <span m="969270">once.</span> <span m="974110">And</span> <span m="974580">we</span> <span m="974700">can</span> <span m="974870">see</span> <span m="975240">that</span> <span m="977720">the</span> <span m="977780">function is</span> <span m="978180">bijective</span> <span m="978750">if</span> <span m="978910">and</span> <span m="979030">only</span> <span m="979290">if</span> <span m="979430">it</span> <span m="979550">is</span> <span m="979670">both</span> <span m="980120">surjective</span> <span m="980780">and</span> <span m="981040">injective.</span> <span m="982070">So</span> <span m="983260">bijective</span> <span m="985630">if</span> <span m="985860">and</span> <span m="985980">only</span> <span m="986270">if</span> <span m="986940">we</span> <span m="987080">have</span> <span m="987440">both</span> <span m="987760">the</span> <span m="987850">properties</span> <span m="988920">surjectvie</span> <span m="990810">as</span> <span m="991070">well</span> <span m="991460">as</span> <span m="991930">injective.</span> </p>
<p><span m="994550">So</span> <span m="994740">let's</span> <span m="994930">give</span> <span m="995090">a</span> <span m="995180">couple</span> <span m="995510">of</span> <span m="996060">examples.</span> <span m="1004510">So</span> <span m="1004830">as the</span> <span m="1004910">first</span> <span m="1005220">example,</span> <span m="1006540">we</span> <span m="1006620">may</span> <span m="1006870">have</span> <span m="1007170">the</span> <span m="1007270">set</span> <span m="1007560">x</span> <span m="1007980">and</span> <span m="1008260">y.</span> <span m="1009090">We</span> <span m="1009240">have</span> <span m="1009640">1,</span> <span m="1010240">2,</span> <span m="1010690">and</span> <span m="1010900">3,</span> <span m="1011175">and</span> <span m="1011450">set</span> <span m="1011820">it</span> <span m="1011910">to</span> <span m="1012180">elements</span> <span m="1012700">a and</span> <span m="1013040">b</span> <span m="1013220">over</span> <span m="1013480">here.</span> <span m="1014780">1</span> <span m="1015080">is</span> <span m="1015250">mapped</span> <span m="1015590">to</span> <span m="1015910">a,</span> <span m="1016263">2</span> <span m="1016970">is</span> <span m="1017140">mapped</span> <span m="1017360">to</span> <span m="1017710">a,</span> <span m="1018110">and</span> <span m="1018260">3 is</span> <span m="1018630">mapped</span> <span m="1019100">to</span> <span m="1019250">b.</span> </p>
<p><span m="1020330">And</span> <span m="1020340">now</span> <span m="1020440">we</span> <span m="1020550">can</span> <span m="1020700">see</span> <span m="1020900">that</span> <span m="1021030">every</span> <span m="1021330">element</span> <span m="1021820">in</span> <span m="1021950">y</span> <span m="1023020">is</span> <span m="1023210">mapped</span> <span m="1023480">to</span> <span m="1023620">at</span> <span m="1023710">least</span> <span m="1024030">once.</span> <span m="1024440">This</span> <span m="1024630">one</span> <span m="1024750">is</span> <span m="1024859">mapped</span> <span m="1025060">to</span> <span m="1025180">two</span> <span m="1025390">times.</span> <span m="1025800">This</span> <span m="1025990">one</span> <span m="1026140">is</span> <span m="1026240">mapped</span> <span m="1026470">to</span> <span m="1026680">once.</span> <span m="1027750">So</span> <span m="1027960">this</span> <span m="1028170">one</span> <span m="1028300">is</span> <span m="1028430">actually</span> <span m="1028750">surjective.</span> </p>
<p><span m="1030490">So</span> <span m="1030660">that's</span> <span m="1030890">great.</span> <span m="1032760">Another</span> <span m="1033160">example</span> <span m="1033780">of</span> <span m="1034640">something</span> <span m="1035359">this is</span> <span m="1035730">injective</span> <span m="1036380">is</span> <span m="1036770">if</span> <span m="1036930">you</span> <span m="1037040">have,</span> <span m="1037270">say,</span> <span m="1037550">1,</span> <span m="1037869">2,</span> <span m="1038220">and</span> <span m="1038410">3,</span> <span m="1039319">and</span> <span m="1039650">a,</span> <span m="1040106">b,</span> <span m="1040562">c,</span> <span m="1041020">and</span> <span m="1041270">d.</span> <span m="1042148">1</span> <span m="1042589">is</span> <span m="1042740">mapped,</span> <span m="1043400">say,</span> <span m="1043670">to</span> <span m="1044120">a.</span> <span m="1045020">2</span> <span m="1045470">to b,</span> <span m="1045890">3 to</span> <span m="1046140">d.</span> <span m="1047420">Well,</span> <span m="1047560">in</span> <span m="1047630">this</span> <span m="1047839">case</span> <span m="1048560">we</span> <span m="1048700">have</span> <span m="1049520">that</span> <span m="1049670">it's</span> <span m="1049890">injective.</span> </p>
<p><span m="1050320">Because</span> <span m="1050610">every</span> <span m="1051010">element</span> <span m="1051460">of</span> <span m="1051570">y</span> <span m="1052200">is</span> <span m="1052470">mapped</span> <span m="1052780">to</span> <span m="1053120">at</span> <span m="1053330">most</span> <span m="1053710">once,</span> <span m="1054614">once,</span> <span m="1055066">once,</span> <span m="1055520">zero</span> <span m="1055830">times,</span> <span m="1056540">and</span> <span m="1056700">once.</span> <span m="1057140">So</span> <span m="1057280">this</span> <span m="1057450">one</span> <span m="1057580">is</span> <span m="1057680">injective.</span> <span m="1059340">And</span> <span m="1059860">this</span> <span m="1060020">one</span> <span m="1060190">is</span> <span m="1060300">not</span> <span m="1060490">injective,</span> <span m="1061050">right?</span> </p>
<p><span m="1061300">Because</span> <span m="1061620">this</span> <span m="1061780">one is</span> <span m="1062225">mapped to 2</span> <span m="1062670">times.</span> <span m="1063380">This</span> <span m="1063600">one</span> <span m="1063750">is</span> <span m="1063870">not</span> <span m="1064010">surjective.</span> <span m="1065010">Because</span> <span m="1065300">this</span> <span m="1065480">one</span> <span m="1065640">is</span> <span m="1065750">not</span> <span m="1065890">covered</span> <span m="1066210">at</span> <span m="1066310">all.</span> <span m="1066550">It's</span> <span m="1066660">only</span> <span m="1066850">mapped</span> <span m="1067080">to</span> <span m="1067190">once.</span> </p>
<p><span m="1069250">OK.</span> <span m="1069580">So</span> <span m="1071030">let</span> <span m="1071200">us</span> <span m="1071350">talk again</span> <span m="1071820">about</span> <span m="1072160">permutations.</span> <span m="1074980">So</span> <span m="1075280">let</span> <span m="1075420">me</span> <span m="1075520">talk</span> <span m="1075750">about</span> <span m="1075980">permutations.</span> <span m="1077230">We</span> <span m="1077340">can</span> <span m="1077500">define</span> <span m="1077870">a</span> <span m="1077910">mapping</span> <span m="1078320">using</span> <span m="1078650">a</span> <span m="1078710">permutation</span> <span m="1080770">that</span> <span m="1081040">is an</span> <span m="1081230">example</span> <span m="1081800">of</span> <span m="1082130">a</span> <span m="1082660">bijection.</span> <span m="1083640">So</span> <span m="1083840">let's</span> <span m="1084070">do</span> <span m="1084220">that.</span> </p>
<p><span m="1086290">And</span> <span m="1086420">then</span> <span m="1086520">we</span> <span m="1086600">can</span> <span m="1086760">come</span> <span m="1087060">to</span> <span m="1087240">the</span> <span m="1087650">mapping</span> <span m="1088030">rule.</span> <span m="1089600">And</span> <span m="1089710">we</span> <span m="1089800">can</span> <span m="1089910">start</span> <span m="1090270">to</span> <span m="1090340">do</span> <span m="1090460">some</span> <span m="1090650">counting.</span> <span m="1094240">So</span> <span m="1094490">for</span> <span m="1094680">example</span> <span m="1096680">if</span> <span m="1096830">we</span> <span m="1096990">have</span> <span m="1097690">a</span> <span m="1098490">permutation,</span> <span m="1100690">a</span> <span m="1100890">1</span> <span m="1101620">up</span> <span m="1101820">to</span> <span m="1102080">a</span> <span m="1102300">n,</span> <span m="1104710">so</span> <span m="1104900">let</span> <span m="1105040">this</span> <span m="1105220">be</span> <span m="1105350">a</span> <span m="1105400">permutation</span> <span m="1109040">of</span> <span m="1109290">the</span> <span m="1109400">set</span> <span m="1111270">S</span> <span m="1111640">that</span> <span m="1111830">contains</span> <span m="1112480">all</span> <span m="1112900">the</span> <span m="1113000">elements</span> <span m="1113620">a</span> <span m="1113810">1,</span> <span m="1115080">up to</span> <span m="1115470">a</span> <span m="1115720">n.</span> </p>
<p><span m="1117130">So</span> <span m="1117270">this</span> <span m="1117340">is</span> <span m="1117590">just</span> <span m="1117840">one</span> <span m="1118020">example</span> <span m="1118490">of a</span> <span m="1118600">permutation.</span> <span m="1119930">And</span> <span m="1120110">now</span> <span m="1120240">we</span> <span m="1120370">may</span> <span m="1120560">define</span> <span m="1121170">the</span> <span m="1121280">following</span> <span m="1121740">function.</span> <span m="1122860">We</span> <span m="1122990">say</span> <span m="1123300">that</span> <span m="1124050">pi</span> <span m="1126610">evaluated</span> <span m="1127460">in</span> <span m="1128210">a</span> <span m="1128525">i</span> <span m="1129150">give</span> <span m="1129370">us</span> <span m="1129600">output</span> <span m="1130490">i.</span> <span m="1132630">Actually,</span> <span m="1132970">what</span> <span m="1133110">I</span> <span m="1133210">mean</span> <span m="1133420">here</span> <span m="1133780">is that</span> <span m="1135760">if</span> <span m="1135970">you</span> <span m="1136080">take</span> <span m="1136410">an</span> <span m="1136520">element</span> <span m="1137030">in</span> <span m="1137260">S,</span> <span m="1139010">then</span> <span m="1139850">this</span> <span m="1140120">one</span> <span m="1140340">is</span> <span m="1140540">mapped</span> <span m="1140960">to</span> <span m="1141230">under</span> <span m="1141450">this</span> <span m="1142070">function</span> <span m="1145660">to</span> <span m="1145950">i</span> <span m="1146840">if</span> <span m="1147090">and</span> <span m="1147230">only</span> <span m="1147610">if</span> <span m="1148530">a is</span> <span m="1149090">in</span> <span m="1149210">the</span> <span m="1149380">i-th</span> <span m="1149700">position</span> <span m="1150350">in</span> <span m="1150430">this</span> <span m="1150600">permutation.</span> </p>
<p><span m="1152000">So</span> <span m="1152230">if</span> <span m="1152380">and</span> <span m="1152530">only</span> <span m="1152830">if</span> <span m="1154340">today</span> <span m="1154840">is</span> <span m="1155150">in</span> <span m="1157340">i-th</span> <span m="1161340">term</span> <span m="1163840">in</span> <span m="1164000">the</span> <span m="1164100">permutation.</span> <span m="1170530">So</span> <span m="1170710">in</span> <span m="1170760">this</span> <span m="1170980">case,</span> <span m="1172060">we</span> <span m="1172190">know</span> <span m="1172490">that</span> <span m="1173690">pi</span> <span m="1174130">is</span> <span m="1174470">bijective.</span> <span m="1175600">And</span> <span m="1175840">why</span> <span m="1176030">is</span> <span m="1176210">this?</span> <span m="1176930">Well,</span> <span m="1179830">we</span> <span m="1179990">know</span> <span m="1180440">from</span> <span m="1180760">the</span> <span m="1180830">definition</span> <span m="1181300">of</span> <span m="1181400">a</span> <span m="1181530">permutation</span> <span m="1182300">that</span> <span m="1182590">any</span> <span m="1182910">permutation</span> <span m="1183850">is</span> <span m="1183940">a</span> <span m="1184030">sequence</span> <span m="1184345">in</span> <span m="1184660">which</span> <span m="1184920">every</span> <span m="1185250">element</span> <span m="1185650">of</span> <span m="1185810">S</span> <span m="1186230">occurs</span> <span m="1186640">exactly</span> <span m="1187270">once.</span> </p>
<p><span m="1188430">So</span> <span m="1188590">that</span> <span m="1188760">means</span> <span m="1189030">that</span> <span m="1189790">every</span> <span m="1191420">position</span> <span m="1192290">is</span> <span m="1192570">covered</span> <span m="1192900">exactly</span> <span m="1193540">once</span> <span m="1194110">by an</span> <span m="1194370">element</span> <span m="1194760">of</span> <span m="1194910">S.</span> <span m="1195950">And</span> <span m="1196090">that</span> <span m="1196250">is</span> <span m="1196370">exactly</span> <span m="1196790">the</span> <span m="1196900">definition</span> <span m="1197480">over</span> <span m="1197740">here</span> <span m="1198710">which</span> <span m="1198850">says</span> <span m="1199190">that</span> <span m="1199340">every</span> <span m="1199630">element</span> <span m="1200060">in</span> <span m="1200140">the</span> <span m="1200270">range</span> <span m="1201170">is</span> <span m="1204210">mapped</span> <span m="1204490">to</span> <span m="1204680">exactly</span> <span m="1205190">once</span> <span m="1205620">by</span> <span m="1205900">an</span> <span m="1206090">element</span> <span m="1206530">in</span> <span m="1206610">a</span> <span m="1206680">domain.</span> <span m="1207960">So</span> <span m="1208010">this</span> <span m="1208180">is</span> <span m="1208970">an</span> <span m="1209100">example</span> <span m="1209620">of</span> <span m="1209730">a</span> <span m="1209890">bijection.</span> </p>
<p><span m="1213520">OK.</span> <span m="1213840">So</span> <span m="1213990">now</span> <span m="1214550">that</span> <span m="1214710">we have</span> <span m="1215380">defined</span> <span m="1215930">functions</span> <span m="1216570">and</span> <span m="1216710">the</span> <span m="1216790">special</span> <span m="1217160">properties,</span> <span m="1218320">let's</span> <span m="1218540">talk</span> <span m="1218790">about</span> <span m="1219570">the</span> <span m="1219730">mapping</span> <span m="1220100">rule,</span> <span m="1223210">which</span> <span m="1223690">we'll</span> <span m="1224080">do</span> <span m="1224220">over</span> <span m="1224540">here.</span> <span m="1228600">And</span> <span m="1231360">now</span> <span m="1231950">for</span> <span m="1232130">the</span> <span m="1232220">first</span> <span m="1232640">time</span> <span m="1232870">we</span> <span m="1232960">start</span> <span m="1233340">to</span> <span m="1234380">talk</span> <span m="1234670">about</span> <span m="1234970">the</span> <span m="1235060">cardinalities</span> <span m="1235690">of sets</span> <span m="1236210">and</span> <span m="1236280">how</span> <span m="1236440">they're</span> <span m="1236600">related</span> <span m="1237030">to</span> <span m="1237140">one</span> <span m="1237340">another.</span> <span m="1240440">So</span> <span m="1240730">the</span> <span m="1240830">mapping</span> <span m="1241180">rule</span> <span m="1241420">is</span> <span m="1242080">that</span> <span m="1243310">first</span> <span m="1243610">of</span> <span m="1245080">all,</span> <span m="1246910">if</span> <span m="1247140">f</span> <span m="1247290">is</span> <span m="1247440">a</span> <span m="1247500">function</span> <span m="1247830">from</span> <span m="1247990">x</span> <span m="1248180">to</span> <span m="1248320">y</span> <span m="1248610">and</span> <span m="1248840">if</span> <span m="1249020">f</span> <span m="1249240">is</span> <span m="1249410">actually</span> <span m="1249730">surjective,</span> <span m="1252170">well,</span> <span m="1252350">what</span> <span m="1252460">do</span> <span m="1252570">we</span> <span m="1252690">know?</span> </p>
<p><span m="1253900">We</span> <span m="1254000">actually</span> <span m="1254350">know</span> <span m="1255360">that</span> <span m="1255490">the</span> <span m="1255600">cardinality</span> <span m="1256360">of</span> <span m="1257100">the</span> <span m="1257210">number</span> <span m="1257390">of</span> <span m="1257500">elements</span> <span m="1257830">in</span> <span m="1257890">the</span> <span m="1257970">domain</span> <span m="1259140">is</span> <span m="1259340">at</span> <span m="1259450">least</span> <span m="1259940">the</span> <span m="1260020">number</span> <span m="1260260">of</span> <span m="1260550">elements</span> <span m="1261030">in</span> <span m="1261190">the</span> <span m="1261280">range.</span> <span m="1262200">And</span> <span m="1262340">why</span> <span m="1262530">is that?</span> <span m="1262780">If</span> <span m="1262910">you</span> <span m="1263010">look</span> <span m="1263180">at</span> <span m="1263240">a</span> <span m="1263310">definition</span> <span m="1263810">of</span> <span m="1263880">surjectivity,</span> <span m="1265950">we</span> <span m="1266060">know that</span> <span m="1266950">every</span> <span m="1267410">element</span> <span m="1268050">of</span> <span m="1268290">y</span> <span m="1268600">is</span> <span m="1268810">covered</span> <span m="1269370">by</span> <span m="1270060">some</span> <span m="1270330">element</span> <span m="1270690">in</span> <span m="1270800">x</span> <span m="1271030">at</span> <span m="1271140">least</span> <span m="1271550">once.</span> <span m="1271820">And all</span> <span m="1273770">the</span> <span m="1273850">elements</span> <span m="1274890">in</span> <span m="1277180">x and</span> <span m="1277540">mapped</span> <span m="1277910">to</span> <span m="1278290">exactly</span> <span m="1278710">one</span> <span m="1279350">element</span> <span m="1279820">in</span> <span m="1279990">y.</span> </p>
<p><span m="1280630">So</span> <span m="1280840">we</span> <span m="1280960">know</span> <span m="1281470">that</span> <span m="1282340">the</span> <span m="1282470">cardinality</span> <span m="1283200">of</span> <span m="1283350">x</span> <span m="1283600">is</span> <span m="1283740">at</span> <span m="1283850">least</span> <span m="1284380">y.</span> <span m="1284780">Because</span> <span m="1285170">every</span> <span m="1285540">element</span> <span m="1285890">in</span> <span m="1285950">y is</span> <span m="1286280">mapped</span> <span m="1286680">by</span> <span m="1286890">some</span> <span m="1287330">unique</span> <span m="1287870">distinct</span> <span m="1288350">element</span> <span m="1288810">in</span> <span m="1288950">x.</span> <span m="1290810">And</span> <span m="1292830">if</span> <span m="1293740">a</span> <span m="1293860">function</span> <span m="1294340">f</span> <span m="1294840">is</span> <span m="1295200">injective,</span> <span m="1297950">well,</span> <span m="1298220">in</span> <span m="1298330">that</span> <span m="1298520">case</span> <span m="1298910">we</span> <span m="1299060">know</span> <span m="1299490">that</span> <span m="1302000">the</span> <span m="1302190">reverse</span> <span m="1302550">relation</span> <span m="1302990">holds,</span> <span m="1303990">so</span> <span m="1305190">inequality</span> <span m="1305740">holds.</span> </p>
<p><span m="1306940">The</span> <span m="1307080">cardinality</span> <span m="1307700">of x</span> <span m="1308140">is at</span> <span m="1308300">most</span> <span m="1308650">the</span> <span m="1308880">cardinality</span> <span m="1309195">of</span> <span m="1309510">y.</span> <span m="1311660">So</span> <span m="1311850">why</span> <span m="1312000">is</span> <span m="1312120">that?</span> <span m="1312470">Well,</span> <span m="1312900">every</span> <span m="1313010">element</span> <span m="1313960">in</span> <span m="1315790">an</span> <span m="1315900">injective</span> <span m="1316180">function,</span> <span m="1316590">right?</span> <span m="1316800">Every</span> <span m="1316970">element</span> <span m="1317870">is</span> <span m="1318060">mapped</span> <span m="1318380">to</span> <span m="1318640">at</span> <span m="1318810">most</span> <span m="1319400">one</span> <span m="1319680">element.</span> </p>
<p><span m="1323700">So</span> <span m="1323810">every</span> <span m="1323970">element</span> <span m="1324400">in</span> <span m="1324580">y</span> <span m="1324950">is</span> <span m="1325100">mapped</span> <span m="1325350">by</span> <span m="1325760">at</span> <span m="1325920">most</span> <span m="1326340">one</span> <span m="1326530">element</span> <span m="1326970">in</span> <span m="1327110">x.</span> <span m="1328430">So</span> <span m="1328680">we</span> <span m="1328790">know</span> <span m="1329380">that</span> <span m="1329840">all the</span> <span m="1330100">elements</span> <span m="1330630">in</span> <span m="1330820">x are</span> <span m="1331730">mapped</span> <span m="1331950">to</span> <span m="1332050">some</span> <span m="1332350">element</span> <span m="1332760">in</span> <span m="1332850">y.</span> <span m="1334540">But</span> <span m="1336160">every</span> <span m="1336230">element</span> <span m="1336650">in</span> <span m="1336720">y</span> <span m="1336900">cannot</span> <span m="1337260">map</span> <span m="1337530">to</span> <span m="1337750">by</span> <span m="1338430">more</span> <span m="1338820">than</span> <span m="1338980">two</span> <span m="1339940">times</span> <span m="1340320">by</span> <span m="1340470">something</span> <span m="1341130">in</span> <span m="1341270">the</span> <span m="1341350">domain.</span> <span m="1341970">So</span> <span m="1342030">we</span> <span m="1342140">know</span> <span m="1342260">that this</span> <span m="1342520">inequality</span> <span m="1343070">holds.</span> </p>
<p><span m="1345640">OK.</span> <span m="1345880">For</span> <span m="1346130">a</span> <span m="1346180">bijective</span> <span m="1346670">function,</span> <span m="1347960">we</span> <span m="1348100">have</span> <span m="1348550">that</span> <span m="1349230">both</span> <span m="1349500">these</span> <span m="1349690">properties</span> <span m="1350160">hold.</span> <span m="1350500">And</span> <span m="1350590">we</span> <span m="1350700">will</span> <span m="1350840">have an</span> <span m="1351100">equality</span> <span m="1351660">over</span> <span m="1351910">here.</span> <span m="1354570">So</span> <span m="1356840">if</span> <span m="1357020">this</span> <span m="1357230">one</span> <span m="1357410">is</span> <span m="1357540">bijective,</span> <span m="1362490">we</span> <span m="1362660">have</span> <span m="1363030">that the</span> <span m="1363240">cardinalities</span> <span m="1364070">are</span> <span m="1364240">equal</span> <span m="1364510">to</span> <span m="1364620">one</span> <span m="1364790">another.</span> </p>
<p><span m="1365730">And</span> <span m="1365950">this</span> <span m="1366100">is</span> <span m="1366230">also</span> <span m="1366510">called</span> <span m="1366980">the</span> <span m="1367430">bijection</span> <span m="1368340">rule.</span> <span m="1373640">So</span> <span m="1373700">let's</span> <span m="1373940">give</span> <span m="1374090">an</span> <span m="1374200">example</span> <span m="1375740">where</span> <span m="1376200">we</span> <span m="1376340">want</span> <span m="1376600">to</span> <span m="1378690">find</span> <span m="1379030">out</span> <span m="1379420">how</span> <span m="1379560">many</span> <span m="1379860">ways</span> <span m="1380180">there</span> <span m="1380400">are</span> <span m="1380740">to</span> <span m="1382560">select</span> <span m="1383790">12</span> <span m="1384360">doughnuts</span> <span m="1384960">from</span> <span m="1385300">5</span> <span m="1386310">varieties.</span> <span m="1387110">So</span> <span m="1387280">let's</span> <span m="1387460">see</span> <span m="1387580">how</span> <span m="1387710">that would</span> <span m="1387990">work.</span> </p>
<p><span m="1388920">And</span> <span m="1389060">the</span> <span m="1389120">whole</span> <span m="1389330">idea</span> <span m="1389680">is</span> <span m="1389810">that we're</span> <span m="1390100">going</span> <span m="1390400">to</span> <span m="1391990">define</span> <span m="1392530">the</span> <span m="1392660">set</span> <span m="1393060">that</span> <span m="1393300">we</span> <span m="1393370">want</span> <span m="1393540">to</span> <span m="1393690">count,</span> <span m="1394120">which</span> <span m="1394320">is</span> <span m="1394560">all</span> <span m="1394810">these</span> <span m="1395010">possible</span> <span m="1395400">configurations</span> <span m="1396220">of</span> <span m="1396350">doughnuts</span> <span m="1397480">over</span> <span m="1397760">five</span> <span m="1398310">varieties</span> <span m="1398970">of</span> <span m="1399090">flavors.</span> <span m="1403760">And</span> <span m="1403950">then</span> <span m="1404070">we're</span> <span m="1404170">going</span> <span m="1404400">to</span> <span m="1404520">map</span> <span m="1404830">these</span> <span m="1405570">to</span> <span m="1406390">another</span> <span m="1406770">structure</span> <span m="1407390">that</span> <span m="1407570">we</span> <span m="1407650">can</span> <span m="1407830">understand</span> <span m="1408230">a</span> <span m="1408290">little</span> <span m="1408440">be</span> <span m="1408530">better.</span> <span m="1409870">So</span> <span m="1410060">let's</span> <span m="1410250">do</span> <span m="1410390">this.</span> </p>
<p><span m="1414280">So</span> <span m="1414460">as</span> <span m="1414610">an</span> <span m="1414690">example,</span> <span m="1419540">let</span> <span m="1419750">x</span> <span m="1420350">be</span> <span m="1421620">all</span> <span m="1421930">the</span> <span m="1422030">ways</span> <span m="1425580">to</span> <span m="1426460">select,</span> <span m="1429030">say,</span> <span m="1429250">12</span> <span m="1429660">doughnuts</span> <span m="1432150">from</span> <span m="1432610">5</span> <span m="1433270">varieties.</span> <span m="1435820">So</span> <span m="1435990">let's</span> <span m="1436190">give</span> <span m="1436340">an</span> <span m="1436420">example.</span> <span m="1440780">For</span> <span m="1441000">example,</span> <span m="1441610">we</span> <span m="1441660">may</span> <span m="1441950">have</span> <span m="1442290">2</span> <span m="1442500">doughnuts.</span> <span m="1444000">And</span> <span m="1444710">they</span> <span m="1445010">are</span> <span m="1445780">in</span> <span m="1445990">the</span> <span m="1446120">chocolate</span> <span m="1446570">flavored</span> <span m="1447460">basket.</span> <span m="1448250">So</span> <span m="1448510">we</span> <span m="1448580">have</span> <span m="1448750">chocolate.</span> </p>
<p><span m="1453060">And</span> <span m="1453270">suppose</span> <span m="1453610">we</span> <span m="1453700">have</span> <span m="1454000">no</span> <span m="1455890">doughnuts</span> <span m="1456570">in</span> <span m="1456940">the</span> <span m="1457400">lemon</span> <span m="1457720">filled</span> <span m="1459250">version</span> <span m="1459570">of</span> <span m="1459650">a</span> <span m="1459710">doughnut.</span> <span m="1463140">Suppose</span> <span m="1463610">he</span> <span m="1463730">have</span> <span m="1463980">a</span> <span m="1464020">whole</span> <span m="1464240">bunch</span> <span m="1464540">of</span> <span m="1464670">doughnuts,</span> <span m="1467000">say</span> <span m="1467180">6</span> <span m="1467500">of</span> <span m="1467640">those,</span> <span m="1468860">that</span> <span m="1469090">are</span> <span m="1470570">with sugar.</span> <span m="1472590">We</span> <span m="1472780">have</span> <span m="1473030">some</span> <span m="1473380">that</span> <span m="1473530">are glazed,</span> <span m="1474090">say</span> <span m="1474400">2.</span> <span m="1478050">And</span> <span m="1478630">finally,</span> <span m="1479190">we</span> <span m="1479340">have</span> <span m="1479660">just</span> <span m="1480370">a</span> <span m="1480500">couple</span> <span m="1480940">of</span> <span m="1481230">plain</span> <span m="1481810">doughnuts.</span> </p>
<p><span m="1484120">So</span> <span m="1484270">this</span> <span m="1484480">would</span> <span m="1484620">be</span> <span m="1485440">a</span> <span m="1485560">configuration</span> <span m="1486430">that is</span> <span m="1486660">in</span> <span m="1486810">x.</span> <span m="1487190">Because</span> <span m="1487460">we</span> <span m="1487580">have</span> <span m="1487710">1,</span> <span m="1487980">2,</span> <span m="1488240">3,</span> <span m="1488550">4,</span> <span m="1488820">5,</span> <span m="1489440">varieties.</span> <span m="1490490">And</span> <span m="1490840">we</span> <span m="1491060">have</span> <span m="1492490">12</span> <span m="1492870">doughnuts,</span> <span m="1493932">2</span> <span m="1494410">over</span> <span m="1494690">here,</span> <span m="1494970">6</span> <span m="1495300">here,</span> <span m="1495600">2,</span> <span m="1495920">and another</span> <span m="1496370">2,</span> <span m="1497100">12</span> <span m="1497190">doughnuts,</span> <span m="1498290">that</span> <span m="1498480">are</span> <span m="1498560">selected</span> <span m="1499100">from</span> <span m="1499290">these</span> <span m="1499750">5</span> <span m="1500160">varieties.</span> </p>
<p><span m="1502180">Now,</span> <span m="1504470">if</span> <span m="1504680">you are</span> <span m="1504940">going</span> <span m="1505270">to</span> <span m="1506130">try</span> <span m="1506410">to</span> <span m="1506570">represent</span> <span m="1506960">such a</span> <span m="1507420">configuration,</span> <span m="1508320">that's</span> <span m="1508400">usually</span> <span m="1508850">how</span> <span m="1509020">we</span> <span m="1509150">think</span> <span m="1509400">about</span> <span m="1509750">counting,</span> <span m="1511110">then</span> <span m="1511840">we</span> <span m="1512130">may</span> <span m="1512280">map</span> <span m="1512670">this</span> <span m="1512980">to</span> <span m="1513240">a</span> <span m="1513340">01</span> <span m="1513700">sequence.</span> <span m="1514460">So</span> <span m="1514470">how</span> <span m="1514610">do</span> <span m="1514700">we</span> <span m="1514790">do</span> <span m="1514930">this?</span> <span m="1516040">We</span> <span m="1516100">can</span> <span m="1516230">just</span> <span m="1516460">map</span> <span m="1516700">the</span> <span m="1516790">doughnuts</span> <span m="1517160">two</span> <span m="1517260">0s,</span> <span m="1519190">the</span> <span m="1519570">divider</span> <span m="1519700">between</span> <span m="1520030">the</span> <span m="1520130">two</span> <span m="1520290">baskets</span> <span m="1521560">as</span> <span m="1521770">a</span> <span m="1521850">1.</span> <span m="1523240">So</span> <span m="1523470">this</span> <span m="1523670">is</span> <span m="1523810">a</span> <span m="1523900">1.</span> </p>
<p><span m="1524160">Then</span> <span m="1524270">we</span> <span m="1524400">have</span> <span m="1525320">no</span> <span m="1527060">0</span> <span m="1527890">between</span> <span m="1528350">these</span> <span m="1528600">two</span> <span m="1528780">ones,</span> <span m="1529030">because</span> <span m="1529290">there</span> <span m="1529410">are</span> <span m="1529450">no</span> <span m="1529620">doughnuts</span> <span m="1530030">in</span> <span m="1530180">the</span> <span m="1530350">lemon filled</span> <span m="1530920">basket.</span> <span m="1531540">So</span> <span m="1531710">we</span> <span m="1531800">have</span> <span m="1532080">one</span> <span m="1533040">that</span> <span m="1534580">is</span> <span m="1535130">the</span> <span m="1535170">mapping</span> <span m="1535600">from</span> <span m="1535880">this</span> <span m="1536195">divider</span> <span m="1536510">field</span> <span m="1537140">over</span> <span m="1537460">here.</span> <span m="1538260">We've</span> <span m="1538570">got</span> <span m="1538770">6</span> <span m="1539150">0s.</span> </p>
<p><span m="1541872">We</span> <span m="1542310">have</span> <span m="1542540">another</span> <span m="1542890">one</span> <span m="1543280">that</span> <span m="1543440">is</span> <span m="1543560">this divider,</span> <span m="1544880">two</span> <span m="1545060">0s,</span> <span m="1545630">two</span> <span m="1545800">doughnuts</span> <span m="1546170">in the</span> <span m="1546540">glazed</span> <span m="1546990">version,</span> <span m="1548870">and</span> <span m="1549060">so</span> <span m="1549260">on.</span> <span m="1550250">So</span> <span m="1550460">what</span> <span m="1550550">do</span> <span m="1550640">we</span> <span m="1550760">see</span> <span m="1551070">here?</span> <span m="1551800">We</span> <span m="1552000">have</span> <span m="1552910">a</span> <span m="1553270">01</span> <span m="1554080">sequence</span> <span m="1555380">where</span> <span m="1555520">we</span> <span m="1555660">have</span> <span m="1555910">12</span> <span m="1556240">zeros</span> <span m="1557130">and</span> <span m="1557880">we</span> <span m="1558120">have</span> <span m="1558450">1,</span> <span m="1559010">2,</span> <span m="1559790">3,</span> <span m="1560490">4</span> <span m="1561270">1s.</span> </p>
<p><span m="1563950">And</span> <span m="1564100">we</span> <span m="1564190">can</span> <span m="1564310">see</span> <span m="1564560">that this</span> <span m="1564730">mapping</span> <span m="1565060">is</span> <span m="1565130">actually</span> <span m="1565450">bijective.</span> <span m="1567190">Because</span> <span m="1568230">if</span> <span m="1568440">I</span> <span m="1568550">have</span> <span m="1568830">two</span> <span m="1569050">0s,</span> <span m="1569550">I can</span> <span m="1569650">map</span> <span m="1570020">them</span> <span m="1570430">back</span> <span m="1570650">to</span> <span m="1570790">doughnuts.</span> <span m="1573320">The</span> <span m="1573420">1</span> <span m="1573710">I</span> <span m="1573810">can</span> <span m="1574010">map</span> <span m="1574220">back</span> <span m="1574500">to</span> <span m="1575250">the</span> <span m="1575430">divider</span> <span m="1575690">between</span> <span m="1575950">two</span> <span m="1576270">baskets.</span> <span m="1577710">So</span> <span m="1578070">let</span> <span m="1579600">y</span> <span m="1580050">be</span> <span m="1582790">the</span> <span m="1583000">set</span> <span m="1583660">of</span> <span m="1585510">all</span> <span m="1585730">kinds</span> <span m="1585920">of</span> <span m="1586000">configurations</span> <span m="1586630">of 12,</span> <span m="1587390">all</span> <span m="1587600">kinds</span> <span m="1587810">of</span> <span m="1587880">sequences.</span> </p>
<p><span m="1589100">Oops,</span> <span m="1589340">maybe</span> <span m="1589590">I</span> <span m="1589620">will</span> <span m="1589750">not</span> <span m="1590570">take</span> <span m="1590800">this</span> <span m="1590980">one</span> <span m="1591210">out.</span> <span m="1592815">Let's</span> <span m="1593240">do</span> <span m="1593420">this</span> <span m="1593620">one.</span> <span m="1596840">So</span> <span m="1598390">if</span> <span m="1598580">you are</span> <span m="1598740">going</span> <span m="1598950">to</span> <span m="1599070">define</span> <span m="1599550">y</span> <span m="1600380">as</span> <span m="1600760">the</span> <span m="1600870">set</span> <span m="1607530">of</span> <span m="1608170">all</span> <span m="1610250">16-bit</span> <span m="1611630">sequences</span> <span m="1613910">with</span> <span m="1614170">exactly</span> <span m="1615450">four</span> <span m="1615790">1s,</span> <span m="1622280">then</span> <span m="1622460">we</span> <span m="1622580">know</span> <span m="1623100">that</span> <span m="1623770">by</span> <span m="1624050">the</span> <span m="1624170">bijection</span> <span m="1624630">rule,</span> <span m="1625200">we</span> <span m="1625340">have</span> <span m="1625690">created</span> <span m="1626140">this</span> <span m="1626270">bijection</span> <span m="1626880">over</span> <span m="1627130">here,</span> <span m="1628280">that</span> <span m="1629010">the</span> <span m="1629140">cardinalities</span> <span m="1629990">of</span> <span m="1630100">x</span> <span m="1630360">and</span> <span m="1630520">y</span> <span m="1630810">are</span> <span m="1630970">exactly</span> <span m="1631470">equal.</span> </p>
<p><span m="1632220">So</span> <span m="1632400">now</span> <span m="1632630">we</span> <span m="1632740">know</span> <span m="1633420">that</span> <span m="1633560">by</span> <span m="1633710">the</span> <span m="1633900">bijection</span> <span m="1634340">rule,</span> <span m="1635530">we have</span> <span m="1635780">been able</span> <span m="1636120">to</span> <span m="1636280">count</span> <span m="1638110">the</span> <span m="1638220">number</span> <span m="1638520">of</span> <span m="1638640">elements</span> <span m="1639050">in</span> <span m="1639180">x</span> <span m="1639620">by counting</span> <span m="1640060">something</span> <span m="1640470">else,</span> <span m="1641080">which</span> <span m="1641280">is</span> <span m="1641400">really</span> <span m="1641830">how</span> <span m="1641970">we</span> <span m="1642080">proceed</span> <span m="1642600">in</span> <span m="1642680">these</span> <span m="1642900">types</span> <span m="1643150">of</span> <span m="1643250">proofs.</span> <span m="1644390">So</span> <span m="1647170">we are</span> <span m="1647290">now</span> <span m="1647470">able</span> <span m="1647880">to</span> <span m="1648000">just</span> <span m="1648340">count</span> <span m="1648870">these</span> <span m="1649100">types</span> <span m="1649380">of</span> <span m="1649530">objects.</span> <span m="1650150">And</span> <span m="1650300">later</span> <span m="1650600">on</span> <span m="1651480">next</span> <span m="1651820">lecture,</span> <span m="1652430">we'll</span> <span m="1652550">actually</span> <span m="1653340">figure</span> <span m="1653630">out</span> <span m="1653740">a</span> <span m="1653810">formula</span> <span m="1654290">for</span> <span m="1654400">this</span> <span m="1654600">one.</span> </p>
<p><span m="1657110">So</span> <span m="1657240">this</span> <span m="1657450">is</span> <span m="1657540">an</span> <span m="1657630">example</span> <span m="1658200">of</span> <span m="1658280">how</span> <span m="1658470">we</span> <span m="1658590">can</span> <span m="1658740">use</span> <span m="1658940">the</span> <span m="1658980">bijection</span> <span m="1659500">rule.</span> <span m="1662460">Another</span> <span m="1662850">example</span> <span m="1663570">is</span> <span m="1663850">one</span> <span m="1664180">in</span> <span m="1664370">which</span> <span m="1665770">we are</span> <span m="1665940">going</span> <span m="1666130">to</span> <span m="1666290">count</span> <span m="1666800">subsets.</span> <span m="1672510">So</span> <span m="1672840">we'll</span> <span m="1673020">give a</span> <span m="1673070">lot</span> <span m="1673250">of</span> <span m="1673350">examples</span> <span m="1674210">through</span> <span m="1674450">these</span> <span m="1674920">two</span> <span m="1675060">lectures.</span> <span m="1676140">And</span> <span m="1676320">also</span> <span m="1676520">the</span> <span m="1676620">problem</span> <span m="1676970">set,</span> <span m="1677410">as</span> <span m="1677500">you will</span> <span m="1677780">see,</span> <span m="1678040">will</span> <span m="1678180">have</span> <span m="1678390">a</span> <span m="1678460">lot</span> <span m="1678670">of</span> <span m="1680380">small</span> <span m="1680780">little</span> <span m="1681040">parts</span> <span m="1682260">with</span> <span m="1682420">all</span> <span m="1682610">kinds</span> <span m="1682860">of</span> <span m="1682980">countings</span> <span m="1683440">that you will</span> <span m="1683870">need</span> <span m="1684060">to</span> <span m="1684190">do</span> <span m="1684560">applying</span> <span m="1684950">different</span> <span m="1685330">rules.</span> </p>
<p><span m="1688680">So</span> <span m="1689440">let's</span> <span m="1689870">talk</span> <span m="1690150">about</span> <span m="1691390">how</span> <span m="1691620">to</span> <span m="1691850">count</span> <span m="1692950">subsets</span> <span m="1693800">of</span> <span m="1694085">a</span> <span m="1694370">set</span> <span m="1694590">x.</span> <span m="1695860">So</span> <span m="1696990">what</span> <span m="1697180">we</span> <span m="1697380">want</span> <span m="1697670">is</span> <span m="1697970">a</span> <span m="1698120">bijection</span> <span m="1700480">from</span> <span m="1701910">subsets</span> <span m="1705630">of</span> <span m="1706300">a</span> <span m="1706400">set</span> <span m="1706760">x</span> <span m="1707180">containing</span> <span m="1707555">of,</span> <span m="1707930">say,</span> <span m="1708420">1</span> <span m="1708860">up</span> <span m="1709130">to</span> <span m="1709480">n,</span> <span m="1711160">so</span> <span m="1711310">the</span> <span m="1711390">integers</span> <span m="1711830">1</span> <span m="1712020">up</span> <span m="1712120">to</span> <span m="1712250">n,</span> <span m="1713470">to</span> <span m="1714840">n-bit</span> <span m="1715960">sequences.</span> <span m="1717130">We</span> <span m="1717290">know</span> <span m="1717950">that</span> <span m="1718090">we</span> <span m="1718170">can</span> <span m="1718350">do</span> <span m="1718520">this</span> <span m="1719370">if</span> <span m="1719560">you</span> <span m="1719640">define</span> <span m="1720050">a</span> <span m="1720100">bijection</span> <span m="1720435">as</span> <span m="1720770">follows.</span> </p>
<p><span m="1723800">So</span> <span m="1725510">we</span> <span m="1725690">map</span> <span m="1726640">a</span> <span m="1726750">subset</span> <span m="1727300">S</span> <span m="1728880">under</span> <span m="1730380">a</span> <span m="1730470">mapping</span> <span m="1730880">f</span> <span m="1731720">to</span> <span m="1733080">a</span> <span m="1733170">bit</span> <span m="1733410">sequence,</span> <span m="1734150">b1,</span> <span m="1735100">b1,</span> <span m="1736080">all</span> <span m="1736280">the</span> <span m="1736370">way</span> <span m="1736510">to</span> <span m="1736670">bn.</span> <span m="1738960">If</span> <span m="1739150">I</span> <span m="1739300">add</span> <span m="1739520">the</span> <span m="1739600">following</span> <span m="1740940">relation,</span> <span m="1742340">bi</span> <span m="1742880">is</span> <span m="1743050">computed</span> <span m="1743750">as</span> <span m="1744090">either a</span> <span m="1744440">1</span> <span m="1744700">or a</span> <span m="1744910">0,</span> <span m="1746060">it's</span> <span m="1746250">computed</span> <span m="1746630">as</span> <span m="1746790">a</span> <span m="1746880">1</span> <span m="1747770">if</span> <span m="1749070">i</span> <span m="1749500">is</span> <span m="1749880">in</span> <span m="1750140">S.</span> <span m="1751500">And</span> <span m="1751820">it's</span> <span m="1752640">a</span> <span m="1752950">0</span> <span m="1753450">if</span> <span m="1753710">i</span> <span m="1754190">is not</span> <span m="1754610">in</span> <span m="1754790">S.</span> </p>
<p><span m="1756320">Now,</span> <span m="1756440">we</span> <span m="1756540">know that</span> <span m="1756810">this is</span> <span m="1756930">a</span> <span m="1757160">bijection.</span> <span m="1758680">So</span> <span m="1759230">if</span> <span m="1759400">we</span> <span m="1759510">have</span> <span m="1759900">a</span> <span m="1760030">bit</span> <span m="1760260">sequence,</span> <span m="1760970">then we</span> <span m="1761330">can</span> <span m="1761660">construct</span> <span m="1762440">from</span> <span m="1762700">this</span> <span m="1762900">mapping</span> <span m="1764010">the</span> <span m="1764130">corresponding</span> <span m="1764780">subset.</span> <span m="1765880">If</span> <span m="1766040">you</span> <span m="1766140">have</span> <span m="1766270">a</span> <span m="1766330">subset,</span> <span m="1767410">we</span> <span m="1767510">can</span> <span m="1767840">use</span> <span m="1768060">this</span> <span m="1768200">mapping</span> <span m="1768620">to</span> <span m="1769030">construct</span> <span m="1769520">the</span> <span m="1771010">corresponding</span> <span m="1771500">bit</span> <span m="1771690">sequence.</span> </p>
<p><span m="1778840">So</span> <span m="1779780">how</span> <span m="1779990">many</span> <span m="1781780">n-bit</span> <span m="1782870">sequences</span> <span m="1783530">are there?</span> <span m="1788190">Well,</span> <span m="1789260">there</span> <span m="1789470">are</span> <span m="1789750">2</span> <span m="1789960">to</span> <span m="1790080">the</span> <span m="1790190">power</span> <span m="1790680">n</span> <span m="1791850">n-bit</span> <span m="1792280">sequences.</span> <span m="1792790">Why is</span> <span m="1793160">that?</span> <span m="1793570">Well,</span> <span m="1794170">we</span> <span m="1794270">have</span> <span m="1794470">two</span> <span m="1794690">choices</span> <span m="1795190">for</span> <span m="1795370">b1,</span> <span m="1795580">a</span> <span m="1796020">0</span> <span m="1796350">or</span> <span m="1796500">a</span> <span m="1796570">1,</span> <span m="1797100">two</span> <span m="1797290">choices</span> <span m="1797790">for</span> <span m="1798000">b2,</span> <span m="1798550">0</span> <span m="1798850">or a</span> <span m="1798980">1,</span> <span m="1799740">and</span> <span m="1799930">so</span> <span m="1800070">on.</span> </p>
<p><span m="1800300">So we</span> <span m="1800410">have</span> <span m="1800620">2</span> <span m="1801650">times</span> <span m="1802230">2</span> <span m="1803390">times</span> <span m="1803950">2</span> <span m="1804250">choices</span> <span m="1804990">over</span> <span m="1805320">here.</span> <span m="1807050">So</span> <span m="1807220">we</span> <span m="1807330">have</span> <span m="1807510">2</span> <span m="1807610">to</span> <span m="1807640">the</span> <span m="1807860">power</span> <span m="1808210">n</span> <span m="1808460">choices</span> <span m="1811350">for</span> <span m="1811615">a</span> <span m="1811880">bit</span> <span m="1812120">sequence.</span> <span m="1812710">So</span> <span m="1812840">there are</span> <span m="1813110">2 to the</span> <span m="1813190">power</span> <span m="1813420">of</span> <span m="1813500">n</span> <span m="1814100">number</span> <span m="1814450">of</span> <span m="1814620">bit</span> <span m="1814740">sequences.</span> <span m="1816150">And</span> <span m="1816240">this</span> <span m="1816370">is</span> <span m="1816480">actually</span> <span m="1816830">equal.</span> </p>
<p><span m="1818370">Because</span> <span m="1818730">of</span> <span m="1818840">this</span> <span m="1819620">bijection</span> <span m="1819970">rule</span> <span m="1820320">that we</span> <span m="1820670">have</span> <span m="1820890">described</span> <span m="1821370">over</span> <span m="1821640">here,</span> <span m="1822330">this</span> <span m="1822490">is</span> <span m="1822600">equal</span> <span m="1822980">to</span> <span m="1823100">the</span> <span m="1824520">total</span> <span m="1824780">number</span> <span m="1824990">of</span> <span m="1825090">possible</span> <span m="1825500">ways</span> <span m="1826160">to</span> <span m="1826300">select</span> <span m="1826720">subsets</span> <span m="1827540">of</span> <span m="1827720">x.</span> <span m="1828360">So</span> <span m="1828520">this</span> <span m="1828700">is</span> <span m="1828840">the</span> <span m="1828920">number</span> <span m="1830500">of</span> <span m="1830690">subsets</span> <span m="1833720">of</span> <span m="1835620">an</span> <span m="1835910">n</span> <span m="1837400">element</span> <span m="1840280">set.</span> <span m="1842790">So</span> <span m="1842840">this</span> <span m="1843020">is</span> <span m="1843110">a</span> <span m="1843160">very</span> <span m="1844140">nice</span> <span m="1844620">way</span> <span m="1844850">to</span> <span m="1847060">demonstrate</span> <span m="1847790">how</span> <span m="1848340">we</span> <span m="1848570">can</span> <span m="1848770">use</span> <span m="1849070">a</span> <span m="1849120">bijection</span> <span m="1849700">rule</span> <span m="1849940">to</span> <span m="1850150">count</span> <span m="1850430">something</span> <span m="1850880">that</span> <span m="1851080">appears</span> <span m="1851410">to</span> <span m="1851520">be</span> <span m="1851850">much</span> <span m="1852120">more</span> <span m="1852360">harder</span> <span m="1852800">to</span> <span m="1852940">think</span> <span m="1853230">about,</span> <span m="1853670">to</span> <span m="1853880">grasp.</span> <span m="1854900">At</span> <span m="1855020">least</span> <span m="1855230">for</span> <span m="1855310">me</span> <span m="1855450">it's</span> <span m="1855590">harder</span> <span m="1855830">to</span> <span m="1855900">grasp.</span> </p>
<p><span m="1856320">So</span> <span m="1856540">I have</span> <span m="1856740">a</span> <span m="1856800">subset</span> <span m="1857310">that</span> <span m="1857410">can</span> <span m="1857570">be</span> <span m="1857730">any</span> <span m="1857980">size</span> <span m="1860100">in</span> <span m="1860520">a</span> <span m="1860760">set</span> <span m="1861150">of</span> <span m="1861360">n</span> <span m="1861610">elements.</span> <span m="1862750">And</span> <span m="1862920">now</span> <span m="1863070">I</span> <span m="1863170">can</span> <span m="1863880">find</span> <span m="1864210">this</span> <span m="1864380">really</span> <span m="1864700">easy</span> <span m="1864960">going</span> <span m="1865250">mapping</span> <span m="1865780">that</span> <span m="1865970">I</span> <span m="1866020">can</span> <span m="1866180">show</span> <span m="1866400">to</span> <span m="1866510">be</span> <span m="1866650">bijective</span> <span m="1867140">and</span> <span m="1867260">all of</span> <span m="1867350">a</span> <span m="1867610">sudden,</span> <span m="1868590">I</span> <span m="1868770">know</span> <span m="1869020">how</span> <span m="1869170">to</span> <span m="1869340">count</span> <span m="1869620">it.</span> <span m="1869930">Because</span> <span m="1870680">I</span> <span m="1870810">can</span> <span m="1870990">just</span> <span m="1871300">look</span> <span m="1871740">at</span> <span m="1872560">the</span> <span m="1872770">image</span> <span m="1873430">and</span> <span m="1873620">count</span> <span m="1874280">those</span> <span m="1874640">types</span> <span m="1875070">of</span> <span m="1875420">objects,</span> <span m="1876040">in</span> <span m="1876140">this</span> <span m="1876350">case</span> <span m="1876610">n-bit</span> <span m="1876930">sequences.</span> </p>
<p><span m="1877820">I</span> <span m="1877920">get a</span> <span m="1878200">really</span> <span m="1878530">easygoing</span> <span m="1879090">number</span> <span m="1879510">that I</span> <span m="1879780">can</span> <span m="1879920">compute</span> <span m="1880320">fairly</span> <span m="1880750">easily.</span> <span m="1881730">And</span> <span m="1881850">now</span> <span m="1882020">I</span> <span m="1882150">have</span> <span m="1883630">counted</span> <span m="1883950">something</span> <span m="1884280">much</span> <span m="1884450">more</span> <span m="1884610">complex.</span> <span m="1885800">So</span> <span m="1885840">this</span> <span m="1886030">is</span> <span m="1886160">how</span> <span m="1886320">we</span> <span m="1886420">generally</span> <span m="1887440">will</span> <span m="1888050">think</span> <span m="1888350">about</span> <span m="1888650">these</span> <span m="1888860">things.</span> </p>
<p><span m="1890470">OK.</span> <span m="1890840">So</span> <span m="1891130">let's</span> <span m="1892490">talk</span> <span m="1892850">now</span> <span m="1893160">about</span> <span m="1894800">the</span> <span m="1894910">generalized</span> <span m="1895600">pigeon hole</span> <span m="1896030">principal.</span> <span m="1896410">So</span> <span m="1896540">we have</span> <span m="1896780">covered</span> <span m="1897090">quite</span> <span m="1897330">a</span> <span m="1897370">lot</span> <span m="1897670">of</span> <span m="1898560">definitions</span> <span m="1899790">right</span> <span m="1900090">now.</span> <span m="1902100">So</span> <span m="1902380">we</span> <span m="1902490">explained</span> <span m="1903310">the</span> <span m="1903710">functions</span> <span m="1904160">mapping</span> <span m="1904520">rule.</span> </p>
<p><span m="1904820">So</span> <span m="1904830">now</span> <span m="1904970">we</span> <span m="1905110">come</span> <span m="1905330">to</span> <span m="1905530">generalized</span> <span m="1905960">pigeon</span> <span m="1906250">hole</span> <span m="1906450">principle</span> <span m="1907750">and</span> <span m="1908300">a</span> <span m="1908380">few</span> <span m="1908670">other</span> <span m="1908860">rules.</span> <span m="1910840">OK.</span> <span m="1924820">So</span> <span m="1925030">what</span> <span m="1925170">about</span> <span m="1925440">a</span> <span m="1925480">generalized</span> <span m="1925950">pigeon hole</span> <span m="1926350">principal?</span> <span m="1937940">This</span> <span m="1938130">is</span> <span m="1938290">actually</span> <span m="1942110">the</span> <span m="1942210">following</span> <span m="1942640">counting</span> <span m="1943060">argument.</span> </p>
<p><span m="1944210">If</span> <span m="1944750">I</span> <span m="1944900">know</span> <span m="1945560">that</span> <span m="1946820">the</span> <span m="1946940">cardinality</span> <span m="1947600">of a</span> <span m="1947780">set</span> <span m="1948090">x</span> <span m="1949160">is</span> <span m="1949460">more</span> <span m="1949850">than</span> <span m="1950540">k</span> <span m="1950830">times</span> <span m="1951330">the</span> <span m="1951420">cardinality</span> <span m="1952080">of a set</span> <span m="1952550">y,</span> <span m="1954550">what</span> <span m="1954660">do</span> <span m="1954780">I</span> <span m="1954850">know?</span> <span m="1955670">Well, I</span> <span m="1955780">know</span> <span m="1956000">that</span> <span m="1956140">for</span> <span m="1956530">all</span> <span m="1957140">functions</span> <span m="1957700">f</span> <span m="1958990">that</span> <span m="1959140">have</span> <span m="1959500">domain</span> <span m="1959920">x</span> <span m="1960580">and</span> <span m="1960880">range</span> <span m="1961200">y,</span> <span m="1963770">I</span> <span m="1963880">know</span> <span m="1964060">that there</span> <span m="1964180">must</span> <span m="1964490">exist</span> <span m="1966000">k</span> <span m="1966240">plus</span> <span m="1966600">1</span> <span m="1976400">different</span> <span m="1976710">elements</span> <span m="1979380">of</span> <span m="1979580">x</span> <span m="1981610">that</span> <span m="1981790">are</span> <span m="1981910">mapped</span> <span m="1983700">to</span> <span m="1983730">the</span> <span m="1983850">same</span> <span m="1984200">element</span> <span m="1985000">in</span> <span m="1985200">y.</span> <span m="1995850">And</span> <span m="1996740">if</span> <span m="1996950">we</span> <span m="1997480">take</span> <span m="1997710">a</span> <span m="1997760">specific</span> <span m="1998300">case</span> <span m="1999840">k=1,</span> <span m="2001900">we</span> <span m="2002090">will</span> <span m="2002240">actually</span> <span m="2002580">call</span> <span m="2002810">this</span> <span m="2002960">the</span> <span m="2003040">pigeon</span> <span m="2003300">hole</span> <span m="2003560">principle.</span> </p>
<p><span m="2007930">And let me</span> <span m="2008280">just</span> <span m="2008980">demonstrate</span> <span m="2009415">it</span> <span m="2009850">by</span> <span m="2010110">the</span> <span m="2010690">famous</span> <span m="2011290">example</span> <span m="2011970">of</span> <span m="2012250">pigeons.</span> <span m="2015270">Well,</span> <span m="2015480">if</span> <span m="2015680">I</span> <span m="2015810">have</span> <span m="2017350">more</span> <span m="2018200">pigeons</span> <span m="2018710">that the</span> <span m="2018830">number of</span> <span m="2019230">holes</span> <span m="2019620">than</span> <span m="2019810">they</span> <span m="2020200">can</span> <span m="2020350">fly</span> <span m="2020680">into,</span> <span m="2022010">I</span> <span m="2022130">know</span> <span m="2022350">for</span> <span m="2022480">sure</span> <span m="2022830">there</span> <span m="2023110">exists</span> <span m="2023560">a</span> <span m="2023610">hole</span> <span m="2025120">that</span> <span m="2026100">two</span> <span m="2026480">pigeons</span> <span m="2027190">will</span> <span m="2027600">fit in.</span> <span m="2028510">So</span> <span m="2030480">that's</span> <span m="2030760">where the</span> <span m="2030950">name</span> <span m="2031200">comes</span> <span m="2031510">from.</span> </p>
<p><span m="2033050">So</span> <span m="2033230">let</span> <span m="2033330">me</span> <span m="2033440">write</span> <span m="2033650">it</span> <span m="2033790">out.</span> <span m="2035290">So</span> <span m="2035590">an</span> <span m="2035680">example</span> <span m="2036260">is</span> <span m="2040210">if</span> <span m="2040410">I</span> <span m="2040510">have</span> <span m="2040810">more</span> <span m="2041110">than</span> <span m="2041330">n</span> <span m="2041580">pigeons,</span> <span m="2044230">so</span> <span m="2044690">the pigeons</span> <span m="2045320">from</span> <span m="2045570">the</span> <span m="2045660">set</span> <span m="2045970">x,</span> <span m="2047940">and</span> <span m="2048350">say</span> <span m="2048670">they</span> <span m="2048760">fly</span> <span m="2049580">into</span> <span m="2051590">n</span> <span m="2051830">holes,</span> <span m="2054159">and</span> <span m="2054320">the</span> <span m="2054380">holes</span> <span m="2055060">is</span> <span m="2055260">my</span> <span m="2055420">set</span> <span m="2055730">y,</span> <span m="2057469">well,</span> <span m="2057739">then</span> <span m="2057929">I</span> <span m="2058010">know</span> <span m="2058530">the</span> <span m="2058750">cardinality</span> <span m="2059389">of</span> <span m="2059500">x</span> <span m="2059770">is</span> <span m="2059870">more</span> <span m="2060060">than</span> <span m="2060199">the</span> <span m="2060280">cardinality</span> <span m="2060870">of</span> <span m="2060960">y.</span> <span m="2061520">I</span> <span m="2061639">have</span> <span m="2061909">more</span> <span m="2062170">pigeons</span> <span m="2062699">than</span> <span m="2062800">there</span> <span m="2062969">are</span> <span m="2063130">holes.</span> </p>
<p><span m="2064460">So</span> <span m="2064639">I</span> <span m="2064719">know</span> <span m="2064900">that</span> <span m="2065179">at</span> <span m="2065350">least</span> <span m="2068780">two</span> <span m="2070190">pigeons</span> <span m="2073139">will</span> <span m="2073330">fly</span> <span m="2073790">into</span> <span m="2074110">the</span> <span m="2074219">same</span> <span m="2074510">hole.</span> <span m="2081239">So</span> <span m="2081540">for</span> <span m="2081710">a</span> <span m="2081750">generalized</span> <span m="2082340">case,</span> <span m="2083250">how</span> <span m="2083460">can</span> <span m="2083620">we</span> <span m="2083969">prove</span> <span m="2084370">something</span> <span m="2084699">like</span> <span m="2084929">that?</span> <span m="2085730">Well,</span> <span m="2085929">we</span> <span m="2085940">could</span> <span m="2086179">use,</span> <span m="2086590">for</span> <span m="2086750">example,</span> <span m="2087100">something</span> <span m="2087530">like</span> <span m="2087840">a</span> <span m="2087929">contradiction.</span> </p>
<p><span m="2089350">For</span> <span m="2089469">example,</span> <span m="2091170">suppose</span> <span m="2091780">that</span> <span m="2093330">for</span> <span m="2093510">all</span> <span m="2095480">k</span> <span m="2095659">plus</span> <span m="2095929">1--</span> <span m="2100250">as</span> <span m="2100450">opposed</span> <span m="2100740">to</span> <span m="2100840">the</span> <span m="2100990">negation</span> <span m="2101720">is</span> <span m="2101950">true.</span> <span m="2103120">So</span> <span m="2103620">how do</span> <span m="2103760">we</span> <span m="2103920">prove</span> <span m="2104160">this</span> <span m="2104240">usually?</span> <span m="2104660">So assume</span> <span m="2105120">we</span> <span m="2105240">have</span> <span m="2105460">this.</span> </p>
<p><span m="2106350">We</span> <span m="2106480">want</span> <span m="2106630">to</span> <span m="2106720">prove</span> <span m="2107060">this.</span> <span m="2107430">Well,</span> <span m="2107480">suppose</span> <span m="2107830">that's</span> <span m="2108010">not</span> <span m="2108240">true.</span> <span m="2108430">So</span> <span m="2108750">suppose</span> <span m="2109120">there's</span> <span m="2109300">a</span> <span m="2109340">mapping</span> <span m="2109750">f.</span> <span m="2110900">So</span> <span m="2111010">a set</span> <span m="2111170">for</span> <span m="2111300">all</span> <span m="2111920">k</span> <span m="2112100">plus</span> <span m="2112370">1</span> <span m="2112560">different</span> <span m="2112900">elements</span> <span m="2113280">of</span> <span m="2113430">x,</span> <span m="2114400">well,</span> <span m="2115110">they are</span> <span m="2115820">not</span> <span m="2116580">mapped</span> <span m="2116900">to</span> <span m="2116990">the</span> <span m="2117100">same</span> <span m="2117380">elements</span> <span m="2117820">in</span> <span m="2117930">y.</span> <span m="2118880">But</span> <span m="2119050">what</span> <span m="2119200">I</span> <span m="2119260">really</span> <span m="2119520">know</span> <span m="2119770">then</span> <span m="2120440">is</span> <span m="2120710">that</span> <span m="2121240">every</span> <span m="2121590">element</span> <span m="2122040">in</span> <span m="2122150">y</span> <span m="2122710">is</span> <span m="2122920">mapped</span> <span m="2123310">to</span> <span m="2123730">at</span> <span m="2123980">most</span> <span m="2124470">by</span> <span m="2125120">k</span> <span m="2125580">distinct</span> <span m="2126060">elements</span> <span m="2126500">of</span> <span m="2126670">x.</span> </p>
<p><span m="2127810">So</span> <span m="2127940">that</span> <span m="2128150">means</span> <span m="2128530">that</span> <span m="2129830">the</span> <span m="2131100">total</span> <span m="2131490">number</span> <span m="2131700">of</span> <span m="2131810">elements</span> <span m="2132230">of</span> <span m="2132370">x</span> <span m="2133170">must</span> <span m="2133400">be</span> <span m="2133540">at</span> <span m="2133700">most</span> <span m="2134110">k</span> <span m="2134330">times</span> <span m="2134690">y.</span> <span m="2135320">And</span> <span m="2135500">that's</span> <span m="2135700">not</span> <span m="2135880">true.</span> <span m="2136240">It's</span> <span m="2136410">larger</span> <span m="2136870">by</span> <span m="2137070">assumption.</span> <span m="2138050">So</span> <span m="2138070">it's</span> <span m="2138170">a</span> <span m="2138220">contradiction.</span> </p>
<p><span m="2139690">So</span> <span m="2139840">this</span> <span m="2140020">is a</span> <span m="2140150">very</span> <span m="2140400">general</span> <span m="2141210">principle</span> <span m="2141760">though.</span> <span m="2142570">And</span> <span m="2142760">it's</span> <span m="2142900">worth</span> <span m="2143150">writing</span> <span m="2143640">it</span> <span m="2143720">all</span> <span m="2144030">out.</span> <span m="2144370">Because</span> <span m="2144670">this</span> <span m="2144880">is</span> <span m="2144920">a</span> <span m="2144980">famous</span> <span m="2146560">rule</span> <span m="2147620">that we will</span> <span m="2147810">use</span> <span m="2148860">in</span> <span m="2149670">counting.</span> <span m="2150350">And</span> <span m="2150490">it</span> <span m="2150570">leads</span> <span m="2150840">to</span> <span m="2151250">interestingly</span> <span m="2151860">results.</span> </p>
<p><span m="2153770">OK.</span> <span m="2154160">So</span> <span m="2154700">let's</span> <span m="2155090">give</span> <span m="2155810">another</span> <span m="2156540">example.</span> <span m="2158840">Let's</span> <span m="2159060">think</span> <span m="2159240">about</span> <span m="2159730">Boston.</span> <span m="2160700">In</span> <span m="2160830">Boston,</span> <span m="2161360">we</span> <span m="2161520">have</span> <span m="2161780">say</span> <span m="2162640">a</span> <span m="2162760">half</span> <span m="2163080">a</span> <span m="2163140">million</span> <span m="2163930">non-bald</span> <span m="2164610">people.</span> <span m="2165350">It</span> <span m="2165520">turns</span> <span m="2165880">out</span> <span m="2166830">that</span> <span m="2169190">there</span> <span m="2169460">are</span> <span m="2169630">at</span> <span m="2169760">least</span> <span m="2170240">3</span> <span m="2170520">people</span> <span m="2171810">that</span> <span m="2172450">have</span> <span m="2172900">the exact</span> <span m="2173420">same</span> <span m="2173740">number</span> <span m="2174060">or</span> <span m="2174170">hairs</span> <span m="2174700">on</span> <span m="2174830">the</span> <span m="2174900">head.</span> </p>
<p><span m="2175485">So</span> <span m="2175750">that's</span> <span m="2175960">kind</span> <span m="2176170">of</span> <span m="2176240">weird.</span> <span m="2176560">How</span> <span m="2177030">do</span> <span m="2177130">we</span> <span m="2177230">know</span> <span m="2177430">that?</span> <span m="2177960">I</span> <span m="2178050">cannot</span> <span m="2178450">point</span> <span m="2178780">out</span> <span m="2179700">any</span> <span m="2180320">three</span> <span m="2180930">in</span> <span m="2181090">Boston</span> <span m="2181560">that</span> <span m="2181840">have the</span> <span m="2181930">same</span> <span m="2182180">number</span> <span m="2182420">of</span> <span m="2182540">hairs.</span> <span m="2183520">I</span> <span m="2183800">have</span> <span m="2183920">no</span> <span m="2184150">idea.</span> </p>
<p><span m="2184690">But</span> <span m="2184740">somehow</span> <span m="2185120">I</span> <span m="2185210">can</span> <span m="2185490">count</span> <span m="2186020">and</span> <span m="2186190">use</span> <span m="2186380">this</span> <span m="2186550">principle</span> <span m="2187730">and</span> <span m="2187990">tell</span> <span m="2188240">you</span> <span m="2188460">that</span> <span m="2188840">it</span> <span m="2189020">must</span> <span m="2189380">be</span> <span m="2189550">true</span> <span m="2190060">that</span> <span m="2190560">in</span> <span m="2190780">Boston</span> <span m="2191820">with</span> <span m="2192270">500,000</span> <span m="2193290">people,</span> <span m="2194110">there</span> <span m="2194310">are</span> <span m="2194470">3</span> <span m="2195050">of</span> <span m="2195280">them</span> <span m="2196010">that</span> <span m="2196150">are</span> <span m="2196440">not</span> <span m="2196670">bald.</span> <span m="2196990">So</span> <span m="2197280">we exclude</span> <span m="2197660">the bald</span> <span m="2198040">people.</span> <span m="2198370">Because</span> <span m="2198630">that would be</span> <span m="2198950">easy.</span> <span m="2199270">They</span> <span m="2199600">all have</span> <span m="2199780">0</span> <span m="2200080">hairs.</span> <span m="2200470">But</span> <span m="2200630">say</span> <span m="2200790">non-bald</span> <span m="2201330">people</span> <span m="2202440">that</span> <span m="2204650">actually</span> <span m="2204970">have</span> <span m="2205120">the</span> <span m="2205220">same</span> <span m="2205510">number</span> <span m="2205740">of</span> <span m="2205880">hairs.</span> </p>
<p><span m="2206520">So</span> <span m="2206590">how</span> <span m="2206690">do</span> <span m="2206780">we</span> <span m="2206880">do</span> <span m="2207050">that?</span> <span m="2209170">How</span> <span m="2209230">can</span> <span m="2209400">we</span> <span m="2209650">make</span> <span m="2209970">such</span> <span m="2210220">types</span> <span m="2210480">of</span> <span m="2210600">conclusions?</span> <span m="2212180">So</span> <span m="2212370">say</span> <span m="2212610">Boston</span> <span m="2213060">has</span> <span m="2214380">about</span> <span m="2215150">500,000</span> <span m="2217970">and</span> <span m="2218250">non-bald</span> <span m="2218950">people.</span> <span m="2221860">And</span> <span m="2224010">let's</span> <span m="2224310">call</span> <span m="2224580">this</span> <span m="2225040">set</span> <span m="2225500">x.</span> <span m="2226050">Because</span> <span m="2226270">we're</span> <span m="2226390">going to use</span> <span m="2226850">the</span> <span m="2226900">pigeon hole</span> <span m="2227340">principle.</span> </p>
<p><span m="2229550">So</span> <span m="2229860">our</span> <span m="2230120">claim</span> <span m="2230480">is</span> <span m="2231710">that</span> <span m="2232780">there</span> <span m="2233020">exists</span> <span m="2234990">3</span> <span m="2235250">people</span> <span m="2235910">in</span> <span m="2236100">Boston</span> <span m="2241000">such</span> <span m="2241210">that</span> <span m="2244290">they</span> <span m="2244470">have</span> <span m="2245300">the</span> <span m="2245460">same</span> <span m="2246080">number</span> <span m="2248330">of</span> <span m="2248530">hairs</span> <span m="2250120">on</span> <span m="2250320">their head.</span> <span m="2256030">So</span> <span m="2256690">how</span> <span m="2256850">do</span> <span m="2256930">we</span> <span m="2257050">do</span> <span m="2257200">this?</span> <span m="2259630">Well,</span> <span m="2259830">we</span> <span m="2259950">know</span> <span m="2261320">that</span> <span m="2263640">we</span> <span m="2263780">may</span> <span m="2264440">generally</span> <span m="2264830">assume</span> <span m="2265290">that</span> <span m="2265880">any</span> <span m="2266130">person</span> <span m="2266580">has</span> <span m="2266780">at</span> <span m="2266890">most</span> <span m="2267160">200,000</span> <span m="2268270">hairs</span> <span m="2269160">on</span> <span m="2269320">their</span> <span m="2269430">head.</span> </p>
<p><span m="2270750">So</span> <span m="2272312">the</span> <span m="2272750">number</span> <span m="2273130">of</span> <span m="2273350">hairs</span> <span m="2278010">on</span> <span m="2278330">a head</span> <span m="2280298">is at</span> <span m="2280790">most</span> <span m="2282310">200,000.</span> <span m="2286630">So</span> <span m="2286890">how</span> <span m="2287070">should</span> <span m="2287340">I</span> <span m="2287400">define</span> <span m="2287930">my</span> <span m="2288100">set</span> <span m="2288770">y</span> <span m="2289410">in</span> <span m="2289590">order</span> <span m="2289780">to</span> <span m="2289890">apply</span> <span m="2290280">this</span> <span m="2290540">pigeon</span> <span m="2290820">hole</span> <span m="2291030">principle?</span> <span m="2292080">So</span> <span m="2292240">what</span> <span m="2292370">do</span> <span m="2292450">we</span> <span m="2292550">do?</span> </p>
<p><span m="2295210">So</span> <span m="2295920">I</span> <span m="2296000">want</span> <span m="2296190">to</span> <span m="2296300">have</span> <span m="2297430">mapping,</span> <span m="2298220">right,</span> <span m="2298670">from</span> <span m="2301980">all</span> <span m="2302220">the</span> <span m="2302310">people</span> <span m="2305610">the</span> <span m="2305720">set</span> <span m="2306030">x</span> <span m="2306820">to</span> <span m="2306940">the</span> <span m="2307140">number</span> <span m="2307420">of</span> <span m="2307530">hairs.</span> <span m="2309030">So</span> <span m="2309300">the</span> <span m="2309410">number</span> <span m="2309710">of</span> <span m="2309800">hairs</span> <span m="2310970">on</span> <span m="2311280">one's</span> <span m="2311570">head</span> <span m="2312260">is</span> <span m="2312460">going</span> <span m="2312700">to</span> <span m="2312820">be</span> <span m="2313080">the</span> <span m="2313180">set</span> <span m="2314920">y.</span> <span m="2316440">And</span> <span m="2317940">what</span> <span m="2318100">do</span> <span m="2318190">we</span> <span m="2318330">know?</span> </p>
<p><span m="2318950">We</span> <span m="2319000">know</span> <span m="2319190">that</span> <span m="2319340">the</span> <span m="2319420">cardinality</span> <span m="2320060">of</span> <span m="2320260">y</span> <span m="2320590">is</span> <span m="2321820">at</span> <span m="2321960">most</span> <span m="2322230">200,000.</span> <span m="2323270">Actually,</span> <span m="2324200">the</span> <span m="2324570">way we</span> <span m="2324850">defined</span> <span m="2325270">it</span> <span m="2325730">it's</span> <span m="2325910">exactly</span> <span m="2326340">200,000.</span> <span m="2328350">And</span> <span m="2330380">the</span> <span m="2330620">set</span> <span m="2330950">x</span> <span m="2331780">has a</span> <span m="2332010">cardinality</span> <span m="2332700">of</span> <span m="2332820">about</span> <span m="2333150">500,000.</span> <span m="2334950">So</span> <span m="2335120">what</span> <span m="2335250">do</span> <span m="2335350">we</span> <span m="2335470">know?</span> </p>
<p><span m="2337190">We</span> <span m="2337300">can</span> <span m="2337460">apply</span> <span m="2337840">our</span> <span m="2338090">generalized</span> <span m="2338760">pigeon hole</span> <span m="2339160">principle.</span> <span m="2340060">It's</span> <span m="2340270">very</span> <span m="2340430">surprising,</span> <span m="2341100">because</span> <span m="2342090">we</span> <span m="2342200">notice</span> <span m="2343220">that</span> <span m="2343550">x</span> <span m="2344390">is</span> <span m="2344870">more</span> <span m="2345260">than</span> <span m="2345600">two</span> <span m="2345880">times</span> <span m="2347470">the</span> <span m="2347570">cardinality</span> <span m="2348210">of</span> <span m="2348300">y.</span> <span m="2349530">2</span> <span m="2349730">times</span> <span m="2350170">200,000</span> <span m="2350770">is</span> <span m="2350900">less</span> <span m="2351160">than</span> <span m="2351240">500,000.</span> <span m="2353080">So</span> <span m="2353250">I</span> <span m="2353340">know</span> <span m="2353540">that</span> <span m="2353740">by</span> <span m="2353920">this</span> <span m="2354170">particular</span> <span m="2355570">principle,</span> <span m="2358470">this</span> <span m="2358740">particular</span> <span m="2359190">mapping</span> <span m="2360420">must</span> <span m="2360730">have</span> <span m="2360880">the</span> <span m="2360990">property,</span> <span m="2361420">because</span> <span m="2361670">this</span> <span m="2361790">holds</span> <span m="2362010">for</span> <span m="2362110">all</span> <span m="2362280">mappings,</span> <span m="2363510">that</span> <span m="2363820">there</span> <span m="2364050">are</span> <span m="2364340">at</span> <span m="2364470">least</span> <span m="2364800">k</span> <span m="2365000">plus</span> <span m="2365280">1,</span> <span m="2365830">2</span> <span m="2366020">plus</span> <span m="2366290">1,</span> <span m="2366560">3</span> <span m="2366900">different</span> <span m="2367580">people</span> <span m="2370900">in</span> <span m="2371000">Boston</span> <span m="2371730">out</span> <span m="2371920">of</span> <span m="2372040">the</span> <span m="2372130">set</span> <span m="2372450">x</span> <span m="2373120">that</span> <span m="2373230">are</span> <span m="2373350">mapped</span> <span m="2373680">to</span> <span m="2373790">the</span> <span m="2373910">same</span> <span m="2374310">element</span> <span m="2375940">in</span> <span m="2376150">y.</span> </p>
<p><span m="2376450">That</span> <span m="2376640">means</span> <span m="2376990">that</span> <span m="2377120">they</span> <span m="2377220">have</span> <span m="2377420">the</span> <span m="2377500">same</span> <span m="2377750">number</span> <span m="2377970">of</span> <span m="2378090">hairs</span> <span m="2378410">on</span> <span m="2378480">their</span> <span m="2378600">head.</span> <span m="2379740">So</span> <span m="2379870">this</span> <span m="2380030">is</span> <span m="2380180">kind</span> <span m="2380360">of</span> <span m="2380450">really</span> <span m="2380650">surprising.</span> <span m="2381210">We</span> <span m="2381340">can</span> <span m="2381810">make</span> <span m="2382080">a</span> <span m="2382120">statement</span> <span m="2383270">without</span> <span m="2383690">really</span> <span m="2384080">inspecting</span> <span m="2385080">every</span> <span m="2385390">single</span> <span m="2385780">person's</span> <span m="2386380">head.</span> </p>
<p><span m="2387490">But</span> <span m="2387610">we</span> <span m="2387690">can</span> <span m="2387830">still</span> <span m="2388080">make</span> <span m="2388300">a</span> <span m="2388320">statement</span> <span m="2388860">about</span> <span m="2391810">the</span> <span m="2391890">fact</span> <span m="2392170">that</span> <span m="2392270">there</span> <span m="2392420">are</span> <span m="2392510">3</span> <span m="2393230">different</span> <span m="2393640">people</span> <span m="2393910">in</span> <span m="2394010">Boston</span> <span m="2394700">that have the</span> <span m="2394870">exact</span> <span m="2395200">same</span> <span m="2395420">number</span> <span m="2395630">of</span> <span m="2395710">hairs.</span> <span m="2398020">So</span> <span m="2398570">this</span> <span m="2398740">is</span> <span m="2398830">an</span> <span m="2399580">example</span> <span m="2400300">of a</span> <span m="2400430">non-constructive</span> <span m="2401260">proof.</span> <span m="2401770">And</span> <span m="2401910">I</span> <span m="2401930">will</span> <span m="2402130">give</span> <span m="2402220">another</span> <span m="2402520">one.</span> </p>
<p><span m="2403450">And</span> <span m="2403620">it's</span> <span m="2403790">a</span> <span m="2403900">very</span> <span m="2404250">important</span> <span m="2405740">principle.</span> <span m="2407250">There's</span> <span m="2407460">actually</span> <span m="2407760">a</span> <span m="2407910">new</span> <span m="2408080">technique</span> <span m="2408560">that</span> <span m="2408690">you</span> <span m="2408760">haven't</span> <span m="2409080">seen</span> <span m="2409270">before.</span> <span m="2410400">So</span> <span m="2410530">far</span> <span m="2410810">we</span> <span m="2410940">have</span> <span m="2412150">been</span> <span m="2413110">constructively</span> <span m="2413920">proofing</span> <span m="2414340">all</span> <span m="2414510">kinds</span> <span m="2414740">of</span> <span m="2414830">properties</span> <span m="2415380">using</span> <span m="2415720">induction</span> <span m="2416500">mainly.</span> </p>
<p><span m="2420120">And</span> <span m="2420360">this</span> <span m="2420570">is</span> <span m="2420690">what</span> <span m="2420860">is</span> <span m="2420990">called</span> <span m="2421230">a</span> <span m="2421260">non-constructive</span> <span m="2422150">proof.</span> <span m="2422460">Because</span> <span m="2422760">I</span> <span m="2422850">cannot</span> <span m="2423190">give</span> <span m="2423420">a</span> <span m="2423600">specific</span> <span m="2424150">example</span> <span m="2425510">that</span> <span m="2425720">demonstrates</span> <span m="2426630">that</span> <span m="2426790">this</span> <span m="2427010">claim</span> <span m="2427310">is</span> <span m="2427450">true.</span> <span m="2429230">But</span> <span m="2429470">yet,</span> <span m="2429710">I've</span> <span m="2429890">shown</span> <span m="2430280">that it</span> <span m="2430480">is</span> <span m="2430640">true,</span> <span m="2431860">but</span> <span m="2432000">in</span> <span m="2432110">a</span> <span m="2432250">non-constructive</span> <span m="2432960">way</span> <span m="2433850">without</span> <span m="2434200">an</span> <span m="2434290">example.</span> </p>
<p><span m="2436220">OK.</span> <span m="2436710">So</span> <span m="2437770">what</span> <span m="2437940">about</span> <span m="2438440">another</span> <span m="2439130">one?</span> <span m="2444590">For</span> <span m="2444860">example,</span> <span m="2445170">we</span> <span m="2445270">may</span> <span m="2445650">pick</span> <span m="2446530">10</span> <span m="2446810">arbitrary</span> <span m="2447960">two-digit</span> <span m="2448730">numbers.</span> <span m="2450200">So</span> <span m="2450420">pick</span> <span m="2450790">10</span> <span m="2453240">arbitrary</span> <span m="2456210">double-digit</span> <span m="2458330">numbers.</span> </p>
<p><span m="2462750">And</span> <span m="2464230">we</span> <span m="2464370">can</span> <span m="2465310">pick</span> <span m="2465620">any</span> <span m="2467010">sequence</span> <span m="2467410">of</span> <span m="2467520">numbers.</span> <span m="2467805">I'm</span> <span m="2468090">just</span> <span m="2468820">picking</span> <span m="2469120">a few.</span> <span m="2469740">You</span> <span m="2469910">may</span> <span m="2470660">add</span> <span m="2470910">a few,</span> <span m="2471230">too.</span> <span m="2471840">i don't know,</span> <span m="2472180">2,</span> <span m="2472640">7,</span> <span m="2473100">14,</span> <span m="2474940">I don't</span> <span m="2474980">know,</span> <span m="2475850">31,</span> <span m="2476850">25,</span> <span m="2477850">60,</span> <span m="2479850">92,</span> <span m="2483020">and</span> <span m="2483250">so</span> <span m="2483470">on.</span> <span m="2484230">So</span> <span m="2484560">I</span> <span m="2484720">have</span> <span m="2484920">1, 2,</span> <span m="2485348">3,</span> <span m="2485776">4, 5,</span> <span m="2486204">6,</span> <span m="2486632">7,</span> <span m="2488160">8,</span> <span m="2490510">I don't</span> <span m="2490630">know</span> <span m="2491190">9,</span> <span m="2493740">and</span> <span m="2494050">another</span> <span m="2494370">one,</span> <span m="2494680">say,</span> <span m="2495350">91</span> <span m="2495820">or</span> <span m="2495900">something</span> <span m="2496190">like</span> <span m="2496380">that.</span> </p>
<p><span m="2497720">And</span> <span m="2498500">so</span> <span m="2498640">I</span> <span m="2498700">have</span> <span m="2499080">10</span> <span m="2499730">double-digit</span> <span m="2500570">numbers.</span> <span m="2501450">It</span> <span m="2501640">turns</span> <span m="2502050">out</span> <span m="2502250">that</span> <span m="2502500">I can</span> <span m="2502660">show</span> <span m="2502960">to</span> <span m="2503090">you</span> <span m="2503480">that</span> <span m="2503730">there</span> <span m="2503920">are</span> <span m="2504700">two</span> <span m="2505940">subsets</span> <span m="2507340">that</span> <span m="2508500">if</span> <span m="2508840">I</span> <span m="2508860">look</span> <span m="2509090">at</span> <span m="2509180">the</span> <span m="2509270">sum</span> <span m="2509580">of</span> <span m="2509680">their</span> <span m="2509830">elements,</span> <span m="2510560">so</span> <span m="2510960">I</span> <span m="2511040">look</span> <span m="2511250">at</span> <span m="2511300">sum</span> <span m="2511600">of</span> <span m="2511710">the</span> <span m="2511780">elements</span> <span m="2512090">of</span> <span m="2512190">the</span> <span m="2512280">first</span> <span m="2512670">subset</span> <span m="2513550">and I</span> <span m="2513740">look</span> <span m="2513920">at</span> <span m="2514550">the</span> <span m="2514840">sum</span> <span m="2515060">of</span> <span m="2515170">the</span> <span m="2515240">elements</span> <span m="2515670">of</span> <span m="2515820">the</span> <span m="2516220">second</span> <span m="2516850">subset,</span> <span m="2517810">that</span> <span m="2517940">I</span> <span m="2517990">can</span> <span m="2518170">find</span> <span m="2518490">two</span> <span m="2518660">subsets</span> <span m="2519780">that</span> <span m="2519910">have</span> <span m="2520110">an</span> <span m="2520270">equal</span> <span m="2520600">sum.</span> <span m="2522080">Now</span> <span m="2522140">if</span> <span m="2522350">you</span> <span m="2522620">just</span> <span m="2522930">look</span> <span m="2523160">at</span> <span m="2523290">those</span> <span m="2523560">numbers,</span> <span m="2523930">and I've</span> <span m="2524270">now</span> <span m="2524460">picked</span> <span m="2525020">10</span> <span m="2525260">arbitrary</span> <span m="2526220">double-digit</span> <span m="2526860">numbers,</span> <span m="2527790">well,</span> <span m="2527960">usually</span> <span m="2528310">it's</span> <span m="2528520">pretty</span> <span m="2528760">hard</span> <span m="2529100">to</span> <span m="2529200">figure</span> <span m="2529530">out</span> <span m="2529700">whether</span> <span m="2529880">that's</span> <span m="2530100">really</span> <span m="2530370">true</span> <span m="2530620">or</span> <span m="2530730">not.</span> </p>
<p><span m="2531040">Maybe</span> <span m="2531360">I</span> <span m="2531500">have</span> <span m="2532200">been</span> <span m="2532580">selecting</span> <span m="2533110">the</span> <span m="2533190">numbers</span> <span m="2533460">in</span> <span m="2533550">such a</span> <span m="2533830">way that</span> <span m="2534110">it's</span> <span m="2534450">easy</span> <span m="2534760">to</span> <span m="2534890">see.</span> <span m="2536460">I</span> <span m="2536550">mean,</span> <span m="2536820">we</span> <span m="2536830">can</span> <span m="2536960">still</span> <span m="2537330">try</span> <span m="2537550">to</span> <span m="2537700">wrap</span> <span m="2537920">our</span> <span m="2538220">minds</span> <span m="2538840">around</span> <span m="2539320">it</span> <span m="2539480">and</span> <span m="2539620">try</span> <span m="2539820">to</span> <span m="2539980">really</span> <span m="2540210">solve</span> <span m="2540640">this</span> <span m="2541000">constructively</span> <span m="2541810">by</span> <span m="2541920">giving</span> <span m="2542250">an</span> <span m="2542660">example.</span> <span m="2543880">It</span> <span m="2544070">turns out that</span> <span m="2544410">we</span> <span m="2544600">can</span> <span m="2544750">prove</span> <span m="2545070">this</span> <span m="2545200">statement.</span> <span m="2546380">And</span> <span m="2546800">we</span> <span m="2546900">will</span> <span m="2547000">use the</span> <span m="2547320">pigeon</span> <span m="2547640">hole</span> <span m="2547840">principle.</span> </p>
<p><span m="2549880">And</span> <span m="2550390">we</span> <span m="2550510">do</span> <span m="2550610">not</span> <span m="2550780">even</span> <span m="2551060">have</span> <span m="2551300">to</span> <span m="2551560">actually--</span> <span m="2552200">oh, you</span> <span m="2552550">have</span> <span m="2552830">a</span> <span m="2552930">question?</span> </p>
<p><span m="2553731">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2556320">PROFESSOR: Oh,</span> <span m="2556760">sure.</span> <span m="2557170">We</span> <span m="2557250">can</span> <span m="2557480">make</span> <span m="2557700">it</span> <span m="2557750">double-digits.</span> <span m="2559110">So</span> <span m="2560170">I</span> <span m="2560300">could</span> <span m="2560460">put</span> <span m="2560570">this</span> <span m="2560710">here.</span> <span m="2560900">It'll be</span> <span m="2561290">4,</span> <span m="2561510">2 of</span> <span m="2561670">I</span> <span m="2561720">want</span> <span m="2561980">to.</span> <span m="2564020">But</span> <span m="2564530">yeah,</span> <span m="2565270">just</span> <span m="2565750">select</span> <span m="2566060">something</span> <span m="2566440">else.</span> <span m="2566650">It doesn't</span> <span m="2566940">really</span> <span m="2567140">matter.</span> </p>
<p><span m="2572260">Yeah.</span> <span m="2572430">So</span> <span m="2572580">what</span> <span m="2572730">we</span> <span m="2572820">are</span> <span m="2572890">going</span> <span m="2573080">to</span> <span m="2573180">show</span> <span m="2573490">now</span> <span m="2573800">is</span> <span m="2574050">that</span> <span m="2574360">through</span> <span m="2574540">the</span> <span m="2574660">pigeon</span> <span m="2574970">hole</span> <span m="2575160">principle,</span> <span m="2575820">we</span> <span m="2575920">can</span> <span m="2575980">prove</span> <span m="2576910">that</span> <span m="2577050">there</span> <span m="2577200">are</span> <span m="2577610">two</span> <span m="2578340">subsets</span> <span m="2579120">that</span> <span m="2579240">have</span> <span m="2579430">the</span> <span m="2579500">same</span> <span m="2579800">sum.</span> <span m="2580830">And</span> <span m="2581230">just</span> <span m="2581450">by</span> <span m="2581600">inspection</span> <span m="2582210">it's</span> <span m="2582350">a</span> <span m="2582400">very</span> <span m="2582680">hard</span> <span m="2582900">problem</span> <span m="2583220">to</span> <span m="2583340">solve.</span> <span m="2585420">So</span> <span m="2586320">I</span> <span m="2586470">did not</span> <span m="2586700">even</span> <span m="2586910">give</span> <span m="2587100">you</span> <span m="2587180">an</span> <span m="2587270">example.</span> </p>
<p><span m="2588380">But</span> <span m="2588790">we</span> <span m="2588850">can</span> <span m="2588990">still</span> <span m="2589210">show</span> <span m="2589520">this.</span> <span m="2590600">So</span> <span m="2590790">how</span> <span m="2590970">can</span> <span m="2591110">we</span> <span m="2591200">go</span> <span m="2591460">ahead</span> <span m="2592335">with</span> <span m="2592600">this?</span> <span m="2592970">So</span> <span m="2593150">let's</span> <span m="2593560">think</span> <span m="2593790">together</span> <span m="2594270">about</span> <span m="2594550">this</span> <span m="2595050">problem.</span> </p>
<p><span m="2596740">So</span> <span m="2598260">I</span> <span m="2598400">want</span> <span m="2598660">to</span> <span m="2598800">choose</span> <span m="2599280">two</span> <span m="2599510">sets</span> <span m="2600200">x</span> <span m="2600530">and</span> <span m="2600740">y.</span> <span m="2602060">And</span> <span m="2603030">somehow,</span> <span m="2603420">I</span> <span m="2603540">want</span> <span m="2603710">to</span> <span m="2604000">have</span> <span m="2604240">a</span> <span m="2604320">mapping,</span> <span m="2605170">right,</span> <span m="2606580">from</span> <span m="2608750">any</span> <span m="2610130">double-digit</span> <span m="2612660">set</span> <span m="2612930">of</span> <span m="2613030">numbers.</span> <span m="2614840">Somehow</span> <span m="2615500">I want to</span> <span m="2616370">map</span> <span m="2616660">that</span> <span m="2616950">to</span> <span m="2617210">sums.</span> <span m="2618020">Because</span> <span m="2618320">that's</span> <span m="2618510">what</span> <span m="2618660">I'm</span> <span m="2618820">interested</span> <span m="2619390">in.</span> </p>
<p><span m="2619680">I'm</span> <span m="2619800">interested</span> <span m="2620060">in</span> <span m="2620320">sums.</span> <span m="2620820">And</span> <span m="2621170">I want</span> <span m="2621520">to</span> <span m="2621640">show</span> <span m="2621940">something</span> <span m="2622830">about</span> <span m="2629230">subsets</span> <span m="2630990">of</span> <span m="2631190">these</span> <span m="2631390">double-digit</span> <span m="2632000">numbers.</span> <span m="2632770">So</span> <span m="2632840">what</span> <span m="2632930">do</span> <span m="2633030">I</span> <span m="2633100">do?</span> </p>
<p><span m="2634790">I</span> <span m="2634890">take</span> <span m="2635140">x</span> <span m="2635730">as</span> <span m="2636160">the</span> <span m="2636720">collection</span> <span m="2641310">of</span> <span m="2641470">subsets</span> <span m="2645670">of</span> <span m="2646180">these</span> <span m="2646490">numbers.</span> <span m="2650370">And</span> <span m="2651510">I</span> <span m="2651700">want</span> <span m="2651870">to</span> <span m="2652170">that</span> <span m="2652330">there</span> <span m="2652540">are</span> <span m="2652600">at</span> <span m="2652680">least</span> <span m="2653070">two</span> <span m="2653260">subsets</span> <span m="2653940">that</span> <span m="2654040">map</span> <span m="2654320">to</span> <span m="2654390">the</span> <span m="2654470">same</span> <span m="2654790">sum.</span> <span m="2655880">So</span> <span m="2656390">let's</span> <span m="2656600">first</span> <span m="2656880">count</span> <span m="2657210">how</span> <span m="2657330">many</span> <span m="2657600">we</span> <span m="2657730">have</span> <span m="2658000">here.</span> </p>
<p><span m="2658890">We</span> <span m="2659020">already</span> <span m="2659320">did</span> <span m="2659540">this.</span> <span m="2659790">We</span> <span m="2659910">made</span> <span m="2660180">a</span> <span m="2660230">mapping</span> <span m="2661190">from</span> <span m="2661900">subsets</span> <span m="2662720">to</span> <span m="2664100">two</span> <span m="2664710">binary</span> <span m="2665230">sequences,</span> <span m="2666240">bit</span> <span m="2666510">sequences.</span> <span m="2667660">In</span> <span m="2667800">this</span> <span m="2668030">case,</span> <span m="2668300">we</span> <span m="2668420">have</span> <span m="2669990">10</span> <span m="2670210">numbers.</span> </p>
<p><span m="2671160">So</span> <span m="2671300">we</span> <span m="2671410">have</span> <span m="2671620">2</span> <span m="2671720">to</span> <span m="2671930">the</span> <span m="2672040">power</span> <span m="2672420">10</span> <span m="2673770">possible</span> <span m="2676120">subsets.</span> <span m="2677660">So</span> <span m="2677800">this</span> <span m="2677990">is</span> <span m="2678090">equal</span> <span m="2678340">to</span> <span m="2678460">1,024.</span> <span m="2681870">Now</span> <span m="2681970">y</span> <span m="2682630">is</span> <span m="2682870">going</span> <span m="2683120">to</span> <span m="2683230">be</span> <span m="2683900">the</span> <span m="2684080">sum</span> <span m="2684810">of</span> <span m="2685620">a</span> <span m="2685740">subset.</span> <span m="2686890">So</span> <span m="2688830">what</span> <span m="2688960">do</span> <span m="2689090">I</span> <span m="2689190">know?</span> </p>
<p><span m="2691520">I</span> <span m="2691690">know</span> <span m="2692070">that</span> <span m="2692430">the</span> <span m="2694260">possible</span> <span m="2694680">sums</span> <span m="2695210">range</span> <span m="2695570">from</span> <span m="2695730">0</span> <span m="2697250">all</span> <span m="2697540">the</span> <span m="2697660">way</span> <span m="2698080">to,</span> <span m="2698510">well,</span> <span m="2698680">what's</span> <span m="2698870">the</span> <span m="2698960">maximum?</span> <span m="2700300">sum</span> <span m="2700630">that</span> <span m="2700770">I</span> <span m="2700820">can</span> <span m="2701050">have</span> <span m="2701830">out of</span> <span m="2705160">a</span> <span m="2705290">subset</span> <span m="2705650">of</span> <span m="2705760">10</span> <span m="2705990">double-digit</span> <span m="2706630">numbers?</span> <span m="2707000">So</span> <span m="2707020">I</span> <span m="2707080">can</span> <span m="2707230">select</span> <span m="2707700">all</span> <span m="2707940">the</span> <span m="2708070">10</span> <span m="2710680">elements</span> <span m="2711220">in</span> <span m="2711430">this</span> <span m="2711570">set.</span> <span m="2713110">And</span> <span m="2713400">they are</span> <span m="2713570">double</span> <span m="2713860">digit.</span> </p>
<p><span m="2714320">So</span> <span m="2714530">at</span> <span m="2714640">most,</span> <span m="2716460">they</span> <span m="2716660">are</span> <span m="2716800">99.</span> <span m="2718410">So</span> <span m="2718700">I</span> <span m="2718790">know</span> <span m="2719270">that</span> <span m="2721320">this</span> <span m="2721700">set</span> <span m="2722370">is</span> <span m="2722570">really</span> <span m="2723650">the</span> <span m="2723780">set</span> <span m="2724690">of</span> <span m="2725060">all</span> <span m="2726850">possible</span> <span m="2727790">sums.</span> <span m="2731630">Now</span> <span m="2731800">we</span> <span m="2731910">know that</span> <span m="2732330">1,024</span> <span m="2733230">is</span> <span m="2733670">more than</span> <span m="2734100">990.</span> <span m="2736250">So</span> <span m="2736640">the</span> <span m="2736750">cardinality</span> <span m="2737400">of</span> <span m="2737550">x</span> <span m="2738110">is</span> <span m="2738310">more</span> <span m="2739010">than the</span> <span m="2739130">cardinality</span> <span m="2740020">of</span> <span m="2740140">y.</span> </p>
<p><span m="2740980">So</span> <span m="2741140">by</span> <span m="2741270">the pigeon</span> <span m="2741660">hole</span> <span m="2741860">principle,</span> <span m="2742440">we</span> <span m="2742460">know</span> <span m="2742710">that</span> <span m="2744260">there</span> <span m="2744430">exists</span> <span m="2744850">at</span> <span m="2744960">least</span> <span m="2745240">two</span> <span m="2745430">different</span> <span m="2745960">elements</span> <span m="2746420">of</span> <span m="2746580">x.</span> <span m="2746960">In</span> <span m="2747100">our</span> <span m="2747340">case,</span> <span m="2747700">there</span> <span m="2747930">exists</span> <span m="2748120">two</span> <span m="2748410">different</span> <span m="2748830">subsets</span> <span m="2750710">that</span> <span m="2751050">map</span> <span m="2751330">to</span> <span m="2751450">the</span> <span m="2751600">same</span> <span m="2754430">elements</span> <span m="2754910">in</span> <span m="2755010">y,</span> <span m="2755060">the</span> <span m="2755170">same</span> <span m="2755500">sum.</span> <span m="2756600">So</span> <span m="2756790">now</span> <span m="2756970">we</span> <span m="2757110">have</span> <span m="2757220">shown</span> <span m="2758270">that</span> <span m="2759510">even</span> <span m="2759940">though</span> <span m="2760320">we</span> <span m="2762620">have not</span> <span m="2763050">shown</span> <span m="2763490">any</span> <span m="2763890">particular</span> <span m="2764370">example</span> <span m="2764970">that</span> <span m="2765060">demonstrates</span> <span m="2765760">this claim</span> <span m="2766040">that</span> <span m="2766320">there</span> <span m="2766440">are</span> <span m="2766510">two</span> <span m="2766670">different</span> <span m="2767010">subsets</span> <span m="2768000">that</span> <span m="2768620">have</span> <span m="2768950">the</span> <span m="2769010">same</span> <span m="2769250">sum,</span> <span m="2769810">we</span> <span m="2769910">still</span> <span m="2770910">got</span> <span m="2771070">a</span> <span m="2771140">proof</span> <span m="2771640">using</span> <span m="2772010">counting</span> <span m="2772280">that</span> <span m="2772550">this</span> <span m="2772850">is</span> <span m="2773000">true.</span> <span m="2774460">So</span> <span m="2774510">this</span> <span m="2774690">is</span> <span m="2774820">called</span> <span m="2775020">a</span> <span m="2775050">non-constructive</span> <span m="2775810">proof.</span> </p>
<p><span m="2777500">Let</span> <span m="2777720">me</span> <span m="2777850">write</span> <span m="2778110">it</span> <span m="2778250">down.</span> <span m="2785980">And</span> <span m="2786130">this</span> <span m="2786310">is</span> <span m="2787100">a</span> <span m="2787180">great</span> <span m="2789870">way</span> <span m="2790290">of</span> <span m="2790420">proofing</span> <span m="2792200">properties.</span> <span m="2796720">So</span> <span m="2796870">now</span> <span m="2797120">we</span> <span m="2797260">can</span> <span m="2797690">continue</span> <span m="2798540">with</span> <span m="2802430">another</span> <span m="2803350">definition</span> <span m="2805280">where</span> <span m="2805510">we</span> <span m="2805850">look</span> <span m="2806360">at</span> <span m="2807760">another</span> <span m="2808050">property.</span> </p>
<p><span m="2808610">Over</span> <span m="2808880">here,</span> <span m="2809080">we</span> <span m="2809170">talks</span> <span m="2809450">about</span> <span m="2810110">surjectivity,</span> <span m="2811150">injectivity,</span> <span m="2811490">and</span> <span m="2811880">bijectivity.</span> <span m="2814670">And</span> <span m="2817730">now</span> <span m="2817910">we</span> <span m="2818000">will</span> <span m="2818140">talk</span> <span m="2818450">about</span> <span m="2820460">the</span> <span m="2820580">following</span> <span m="2821510">property.</span> <span m="2822630">We</span> <span m="2822730">say</span> <span m="2823020">that</span> <span m="2824050">a</span> <span m="2824530">k</span> <span m="2826140">to</span> <span m="2826500">1</span> <span m="2831010">function</span> <span m="2833270">is</span> <span m="2834980">f</span> <span m="2835560">from</span> <span m="2835940">x</span> <span m="2836950">to</span> <span m="2837380">y</span> <span m="2838470">actually</span> <span m="2838990">maps</span> <span m="2840970">actually</span> <span m="2841920">k</span> <span m="2842240">elements</span> <span m="2851440">of</span> <span m="2851520">x</span> <span m="2853795">to</span> <span m="2854710">every</span> <span m="2854910">element</span> <span m="2859420">of y.</span> </p>
<p><span m="2862190">So</span> <span m="2863810">what</span> <span m="2863930">do</span> <span m="2864010">we</span> <span m="2864140">know?</span> <span m="2865890">Well,</span> <span m="2866120">we</span> <span m="2866300">can</span> <span m="2868320">have</span> <span m="2868580">the</span> <span m="2868650">following</span> <span m="2869070">counting</span> <span m="2869520">rule</span> <span m="2869770">that we</span> <span m="2870050">call the</span> <span m="2870330">division</span> <span m="2870790">rule.</span> <span m="2872200">And</span> <span m="2873400">it</span> <span m="2873580">says</span> <span m="2874050">that</span> <span m="2874330">if</span> <span m="2874670">f</span> <span m="2875140">such</span> <span m="2875400">a</span> <span m="2875610">type</span> <span m="2875890">of</span> <span m="2875980">function,</span> <span m="2877020">so</span> <span m="2877130">if</span> <span m="2877390">f</span> <span m="2878030">is</span> <span m="2880180">k</span> <span m="2881740">to</span> <span m="2882020">1,</span> <span m="2883550">well</span> <span m="2883800">then</span> <span m="2885530">we</span> <span m="2885660">know</span> <span m="2885940">that</span> <span m="2886290">the</span> <span m="2887810">cardinality</span> <span m="2888400">of</span> <span m="2888500">the</span> <span m="2888570">domain</span> <span m="2889340">is</span> <span m="2889540">equal</span> <span m="2890460">to</span> <span m="2891730">k</span> <span m="2892610">times</span> <span m="2893020">the</span> <span m="2893090">cardinality</span> <span m="2893970">y.</span> </p>
<p><span m="2896170">And</span> <span m="2896290">why</span> <span m="2896450">is that?</span> <span m="2896920">Well, exactly</span> <span m="2897790">k</span> <span m="2898160">elements</span> <span m="2898740">of</span> <span m="2898940">x</span> <span m="2899320">map</span> <span m="2899790">to</span> <span m="2900350">each</span> <span m="2900580">element</span> <span m="2900900">of</span> <span m="2900980">y.</span> <span m="2901160">So</span> <span m="2901740">the</span> <span m="2901880">first</span> <span m="2902150">element</span> <span m="2902490">of</span> <span m="2902630">y,</span> <span m="2902960">we</span> <span m="2903070">have</span> <span m="2903240">k</span> <span m="2903480">elements</span> <span m="2903890">of</span> <span m="2904050">x</span> <span m="2904420">mapped</span> <span m="2904630">to</span> <span m="2904830">it.</span> <span m="2905460">The</span> <span m="2905550">second</span> <span m="2905890">element</span> <span m="2906240">of</span> <span m="2906340">y,</span> <span m="2906695">k</span> <span m="2907050">elements</span> <span m="2907490">mapped</span> <span m="2907750">to</span> <span m="2907840">that</span> <span m="2908080">one.</span> </p>
<p><span m="2908680">So</span> <span m="2908860">we</span> <span m="2908970">know</span> <span m="2909170">that</span> <span m="2909680">the</span> <span m="2909810">domain</span> <span m="2910320">is</span> <span m="2910480">exactly</span> <span m="2911030">k</span> <span m="2911280">times</span> <span m="2912340">the</span> <span m="2912500">range,</span> <span m="2913200">k</span> <span m="2913390">times</span> <span m="2913790">the</span> <span m="2913860">size</span> <span m="2914170">of</span> <span m="2914270">the</span> <span m="2914370">range.</span> <span m="2915930">Now</span> <span m="2916100">this</span> <span m="2916340">division</span> <span m="2916790">rule</span> <span m="2917250">actually</span> <span m="2917610">generalizes</span> <span m="2921830">the</span> <span m="2922140">bijection</span> <span m="2922670">rule,</span> <span m="2924700">which</span> <span m="2925635">I've</span> <span m="2925930">put</span> <span m="2926020">over</span> <span m="2926190">there,</span> <span m="2926735">the</span> <span m="2927150">bijection</span> <span m="2927680">rule.</span> <span m="2930570">And</span> <span m="2930720">why is</span> <span m="2931100">that?</span> </p>
<p><span m="2933030">Well,</span> <span m="2933230">that's</span> <span m="2933430">because</span> <span m="2936040">a</span> <span m="2936450">function</span> <span m="2936830">is</span> <span m="2936940">a</span> <span m="2937350">bijection</span> <span m="2938740">if</span> <span m="2939010">and</span> <span m="2939130">only</span> <span m="2939520">if</span> <span m="2940600">it</span> <span m="2940770">is</span> <span m="2940960">actually</span> <span m="2941490">1</span> <span m="2942930">to</span> <span m="2943410">1.</span> <span m="2945720">So</span> <span m="2945920">if</span> <span m="2946060">you</span> <span m="2946160">replace</span> <span m="2946660">k</span> <span m="2946870">by</span> <span m="2947050">1,</span> <span m="2947900">we</span> <span m="2948070">have</span> <span m="2948440">that</span> <span m="2949820">exactly</span> <span m="2950530">one</span> <span m="2950830">element</span> <span m="2951340">of</span> <span m="2951540">x</span> <span m="2951870">is</span> <span m="2952070">mapped</span> <span m="2952300">to</span> <span m="2952420">every</span> <span m="2952670">element</span> <span m="2953050">in</span> <span m="2953140">y.</span> <span m="2953510">And</span> <span m="2953650">that's</span> <span m="2953980">the</span> <span m="2954070">definition</span> <span m="2954560">of</span> <span m="2954660">a</span> <span m="2954700">bijection.</span> </p>
<p><span m="2957080">And</span> <span m="2957210">the</span> <span m="2957270">bijection</span> <span m="2957800">rule</span> <span m="2957980">says</span> <span m="2958750">that</span> <span m="2958920">if</span> <span m="2959050">you</span> <span m="2959140">have</span> <span m="2959240">a</span> <span m="2959310">bijection,</span> <span m="2960180">then</span> <span m="2960320">the</span> <span m="2960410">cardinality</span> <span m="2961010">of</span> <span m="2961130">the</span> <span m="2961230">domain</span> <span m="2963000">is</span> <span m="2963220">equal</span> <span m="2963450">to</span> <span m="2963550">the</span> <span m="2963640">cardinality</span> <span m="2964260">of</span> <span m="2964510">the</span> <span m="2965070">range,</span> <span m="2965740">so</span> <span m="2965810">for</span> <span m="2965950">k</span> <span m="2966140">equals</span> <span m="2966510">1</span> <span m="2966740">here.</span> <span m="2969350">So</span> <span m="2969620">let's</span> <span m="2969810">give</span> <span m="2969950">an</span> <span m="2970030">example</span> <span m="2970700">on</span> <span m="2970910">how</span> <span m="2971080">this</span> <span m="2971310">works.</span> <span m="2973890">I</span> <span m="2973970">think</span> <span m="2974270">we</span> <span m="2974380">can</span> <span m="2977660">take</span> <span m="2978120">this out</span> <span m="2978530">actually.</span> </p>
<p><span m="2982240">So</span> <span m="2982370">let's</span> <span m="2982570">give</span> <span m="2982710">an</span> <span m="2982780">example</span> <span m="2983270">using</span> <span m="2983760">a</span> <span m="2984380">chessboard,</span> <span m="2986430">where</span> <span m="2986760">we</span> <span m="2986880">have</span> <span m="2987230">2</span> <span m="2987570">identical</span> <span m="2988170">rooks</span> <span m="2989290">and</span> <span m="2990490">we</span> <span m="2990660">want</span> <span m="2990920">to</span> <span m="2991180">count</span> <span m="2991460">the</span> <span m="2991540">number</span> <span m="2991860">of</span> <span m="2992090">ways</span> <span m="2992560">we</span> <span m="2992630">can</span> <span m="2992770">put</span> <span m="2992970">them</span> <span m="2993150">on</span> <span m="2993280">the</span> <span m="2993370">chessboard</span> <span m="2994160">in</span> <span m="2994290">such</span> <span m="2994600">a</span> <span m="2994700">way</span> <span m="2995520">that</span> <span m="2997060">the</span> <span m="2997180">2</span> <span m="2997470">rooks</span> <span m="2997970">cannot</span> <span m="3000430">see</span> <span m="3000610">one</span> <span m="3000850">another,</span> <span m="3001640">meaning</span> <span m="3002020">that</span> <span m="3003410">the</span> <span m="3003540">rooks</span> <span m="3004050">are</span> <span m="3004300">on</span> <span m="3004430">different</span> <span m="3004940">rows</span> <span m="3005350">and</span> <span m="3005560">on</span> <span m="3005710">different</span> <span m="3006040">columns.</span> <span m="3008140">So</span> <span m="3008330">let's</span> <span m="3008540">give</span> <span m="3008700">an</span> <span m="3008780">example.</span> <span m="3010500">So the</span> <span m="3010610">example is</span> <span m="3011170">like</span> <span m="3011340">this.</span> </p>
<p><span m="3011700">So</span> <span m="3011790">how</span> <span m="3012010">many</span> <span m="3012360">ways</span> <span m="3018170">do</span> <span m="3018410">we</span> <span m="3018530">have</span> <span m="3020140">to</span> <span m="3020320">place</span> <span m="3022330">2</span> <span m="3023080">identical</span> <span m="3028010">rooks</span> <span m="3032160">on</span> <span m="3032330">a</span> <span m="3032370">chessboard</span> <span m="3037940">in</span> <span m="3038050">such</span> <span m="3038380">a</span> <span m="3038480">way</span> <span m="3040720">that</span> <span m="3040960">no</span> <span m="3042280">row</span> <span m="3044180">or</span> <span m="3044400">column</span> <span m="3045760">is</span> <span m="3045970">shared?</span> <span m="3050820">So</span> <span m="3051070">how</span> <span m="3051230">can</span> <span m="3051340">we</span> <span m="3051450">do</span> <span m="3051620">this?</span> <span m="3054270">Well,</span> <span m="3056390">for</span> <span m="3056620">example,</span> <span m="3057110">let's</span> <span m="3057470">look</span> <span m="3057780">at</span> <span m="3058120">a</span> <span m="3059420">chessboard.</span> </p>
<p><span m="3064190">And</span> <span m="3065140">suppose</span> <span m="3065580">we</span> <span m="3065680">have</span> <span m="3065860">a</span> <span m="3066050">rook</span> <span m="3066300">over</span> <span m="3066650">here</span> <span m="3067400">and</span> <span m="3067670">a</span> <span m="3067880">rook</span> <span m="3068110">over</span> <span m="3068470">here.</span> <span m="3069980">And</span> <span m="3070410">say</span> <span m="3070610">the</span> <span m="3070720">first</span> <span m="3071130">rook</span> <span m="3072160">is</span> <span m="3072480">on</span> <span m="3072710">row</span> <span m="3073270">1</span> <span m="3074440">and</span> <span m="3074850">on</span> <span m="3076450">column</span> <span m="3076860">1.</span> <span m="3078250">And</span> <span m="3078550">the</span> <span m="3078620">second</span> <span m="3078980">rook</span> <span m="3079890">is</span> <span m="3080210">on</span> <span m="3080650">row</span> <span m="3081210">2,</span> <span m="3083050">R2,</span> <span m="3083720">on</span> <span m="3083820">row</span> <span m="3084040">R2,</span> <span m="3085050">and</span> <span m="3085380">on</span> <span m="3087020">the</span> <span m="3087160">column</span> <span m="3088170">that</span> <span m="3088280">is</span> <span m="3088430">indexed</span> <span m="3088890">by</span> <span m="3089010">C2.</span> </p>
<p><span m="3091280">So</span> <span m="3092410">how</span> <span m="3092560">can</span> <span m="3092730">we</span> <span m="3092790">describe</span> <span m="3093270">such</span> <span m="3093530">configurations?</span> <span m="3095270">Well,</span> <span m="3095560">I</span> <span m="3095650">could</span> <span m="3096770">describe</span> <span m="3097150">this</span> <span m="3097310">by</span> <span m="3097450">using</span> <span m="3097870">a</span> <span m="3097930">sequence</span> <span m="3098830">in</span> <span m="3098980">which</span> <span m="3099290">I</span> <span m="3100230">look</span> <span m="3100470">at</span> <span m="3100550">the</span> <span m="3100650">placement</span> <span m="3101140">of</span> <span m="3101270">the</span> <span m="3101350">first</span> <span m="3101760">rook,</span> <span m="3102230">and</span> <span m="3102480">then</span> <span m="3102670">describe</span> <span m="3103090">the</span> <span m="3103180">placement</span> <span m="3103660">of</span> <span m="3103770">the</span> <span m="3103830">second</span> <span m="3104190">rook.</span> <span m="3104570">So</span> <span m="3104740">I</span> <span m="3104830">may</span> <span m="3105070">have</span> <span m="3105900">r1,</span> <span m="3106380">c1,</span> <span m="3107820">and</span> <span m="3107950">then</span> <span m="3108180">r2</span> <span m="3109180">and</span> <span m="3109390">c2.</span> </p>
<p><span m="3111400">So</span> <span m="3111560">this</span> <span m="3111800">could</span> <span m="3111990">be</span> <span m="3112740">a</span> <span m="3112860">way</span> <span m="3113050">to</span> <span m="3113210">describe</span> <span m="3114680">the</span> <span m="3114830">positioning</span> <span m="3115520">of</span> <span m="3115680">these</span> <span m="3115870">rooks.</span> <span m="3117830">And</span> <span m="3118750">I</span> <span m="3118910">could</span> <span m="3119250">create</span> <span m="3119640">a</span> <span m="3119690">mapping</span> <span m="3120700">f</span> <span m="3121360">that is</span> <span m="3121650">doing</span> <span m="3121880">this</span> <span m="3122080">for</span> <span m="3122250">me.</span> <span m="3123470">And</span> <span m="3125420">so</span> <span m="3125780">if</span> <span m="3125960">I</span> <span m="3126040">call</span> <span m="3126440">this</span> <span m="3127850">an</span> <span m="3127980">example</span> <span m="3129800">of</span> <span m="3130800">a</span> <span m="3130900">valid.</span> <span m="3134020">So let</span> <span m="3134250">y be</span> <span m="3134610">the</span> <span m="3134710">set</span> <span m="3135030">of</span> <span m="3135500">valid</span> <span m="3136180">rook</span> <span m="3136310">configuration.</span> </p>
<p><span m="3137200">And</span> <span m="3137300">this</span> <span m="3137440">is</span> <span m="3137560">one</span> <span m="3137730">example</span> <span m="3138230">of</span> <span m="3138390">it.</span> <span m="3138640">So</span> <span m="3138820">this</span> <span m="3139040">is</span> <span m="3139180">part</span> <span m="3139800">of</span> <span m="3139970">this</span> <span m="3140160">set.</span> <span m="3150380">And</span> <span m="3151550">if</span> <span m="3151810">I</span> <span m="3151900">define</span> <span m="3155100">x</span> <span m="3157520">as</span> <span m="3157790">all the</span> <span m="3157850">sequences</span> <span m="3164510">r1,</span> <span m="3164962">c1,</span> <span m="3166320">r2,</span> <span m="3167120">and</span> <span m="3167300">c2</span> <span m="3168680">such</span> <span m="3169080">that,</span> <span m="3170310">well,</span> <span m="3171900">the</span> <span m="3172220">rook</span> <span m="3172580">over</span> <span m="3172910">here</span> <span m="3173160">does</span> <span m="3173260">not</span> <span m="3173500">share a</span> <span m="3173980">row</span> <span m="3174450">with</span> <span m="3174650">the</span> <span m="3174820">rook</span> <span m="3175490">that</span> <span m="3175670">is</span> <span m="3175940">described</span> <span m="3176360">by</span> <span m="3176470">this</span> <span m="3176620">position.</span> <span m="3177280">So</span> <span m="3177570">r1</span> <span m="3178680">is</span> <span m="3178870">not</span> <span m="3179090">equal</span> <span m="3179380">to</span> <span m="3179570">r2.</span> <span m="3180800">And</span> <span m="3181020">they</span> <span m="3181160">also do</span> <span m="3181410">not</span> <span m="3181580">share</span> <span m="3181920">a</span> <span m="3181990">column.</span> </p>
<p><span m="3182710">So</span> <span m="3185380">the</span> <span m="3185450">first</span> <span m="3185760">rook</span> <span m="3185940">has</span> <span m="3186250">column</span> <span m="3186990">c1.</span> <span m="3187790">The</span> <span m="3187910">second</span> <span m="3188250">one</span> <span m="3188680">is</span> <span m="3189100">on</span> <span m="3189460">column</span> <span m="3189810">c2.</span> <span m="3190860">So</span> <span m="3191100">also</span> <span m="3191410">c1</span> <span m="3192230">and</span> <span m="3192510">c2</span> <span m="3192910">should</span> <span m="3193120">be</span> <span m="3193220">different.</span> <span m="3196080">So</span> <span m="3196290">these</span> <span m="3196470">sequences</span> <span m="3197270">are</span> <span m="3197450">really</span> <span m="3198130">placements,</span> <span m="3198830">right?</span> </p>
<p><span m="3199080">So</span> <span m="3199190">this</span> <span m="3199450">describes</span> <span m="3200280">rook</span> <span m="3202060">1.</span> <span m="3203040">This</span> <span m="3203290">describes</span> <span m="3204150">the</span> <span m="3204240">rook</span> <span m="3205430">2.</span> <span m="3206680">And</span> <span m="3207030">the</span> <span m="3207110">whole</span> <span m="3207310">combination</span> <span m="3208140">is</span> <span m="3208350">really</span> <span m="3211090">a</span> <span m="3211220">placement.</span> </p>
<p><span m="3216950">So</span> <span m="3217130">now</span> <span m="3217280">I have</span> <span m="3217480">described</span> <span m="3218030">the</span> <span m="3218130">function</span> <span m="3218690">f</span> <span m="3219130">that</span> <span m="3219320">maps</span> <span m="3219820">a</span> <span m="3219920">sequence</span> <span m="3221050">that</span> <span m="3221220">describes</span> <span m="3221820">the</span> <span m="3221910">position</span> <span m="3222310">of</span> <span m="3222420">the</span> <span m="3222510">first</span> <span m="3222820">and the</span> <span m="3222970">second</span> <span m="3223880">rook,</span> <span m="3225710">maps</span> <span m="3226600">such</span> <span m="3226840">a</span> <span m="3226890">sequence</span> <span m="3227540">to</span> <span m="3228390">an</span> <span m="3228540">element</span> <span m="3228940">in</span> <span m="3229070">y,</span> <span m="3229450">which</span> <span m="3229550">is</span> <span m="3229650">a</span> <span m="3230080">valid</span> <span m="3230550">rook</span> <span m="3230820">configuration.</span> <span m="3232830">So now</span> <span m="3233230">let's</span> <span m="3233770">have</span> <span m="3233990">a</span> <span m="3234110">look</span> <span m="3234370">at</span> <span m="3234460">how</span> <span m="3234630">we</span> <span m="3234720">can</span> <span m="3234860">apply</span> <span m="3235130">the</span> <span m="3235230">division</span> <span m="3235500">rule.</span> <span m="3236470">So</span> <span m="3238530">is</span> <span m="3238770">this</span> <span m="3238980">function</span> <span m="3240520">bijective?</span> <span m="3242380">Is</span> <span m="3242560">that</span> <span m="3242650">true?</span> </p>
<p><span m="3246770">So</span> <span m="3246920">is</span> <span m="3247030">it true</span> <span m="3247620">that</span> <span m="3249240">every--</span> <span m="3250320">so</span> <span m="3250490">I</span> <span m="3250600">have</span> <span m="3250680">a</span> <span m="3250820">mapping</span> <span m="3251140">that</span> <span m="3251300">goes</span> <span m="3251520">from</span> <span m="3251640">here</span> <span m="3251840">to</span> <span m="3251990">here.</span> <span m="3253530">But</span> <span m="3254520">is</span> <span m="3254680">it true</span> <span m="3254975">that</span> <span m="3255270">every</span> <span m="3255800">valid</span> <span m="3256830">rook</span> <span m="3257060">configuration</span> <span m="3258510">is</span> <span m="3258700">mapped</span> <span m="3259050">to</span> <span m="3259240">exactly</span> <span m="3259770">once?</span> <span m="3262120">Is</span> <span m="3262480">that</span> <span m="3262570">true?</span> </p>
<p><span m="3263870">Is</span> <span m="3264030">this</span> <span m="3269740">the</span> <span m="3269880">only</span> <span m="3270080">sequence</span> <span m="3270410">that</span> <span m="3270740">will</span> <span m="3270860">map</span> <span m="3271260">using</span> <span m="3271630">this</span> <span m="3273180">function</span> <span m="3273520">f</span> <span m="3273960">into</span> <span m="3274270">a valid</span> <span m="3274920">configuration?</span> <span m="3278170">Yup.</span> <span m="3278670">It's</span> <span m="3278910">true.</span> <span m="3280090">So</span> <span m="3281490">you</span> <span m="3281640">can</span> <span m="3281810">switch</span> <span m="3282750">rook</span> <span m="3283010">1</span> <span m="3283250">and rook</span> <span m="3283580">2.</span> <span m="3285140">And</span> <span m="3285480">it will</span> <span m="3285680">still</span> <span m="3286010">look</span> <span m="3286190">the</span> <span m="3286270">same.</span> </p>
<p><span m="3286910">The</span> <span m="3287020">2</span> <span m="3287300">rooks are</span> <span m="3287520">identical.</span> <span m="3288190">They</span> <span m="3288300">look</span> <span m="3288460">exactly</span> <span m="3288860">the</span> <span m="3288960">same.</span> <span m="3289450">So</span> <span m="3289520">we</span> <span m="3289600">have,</span> <span m="3290270">again,</span> <span m="3290730">the exact</span> <span m="3291160">same</span> <span m="3291550">configuration.</span> <span m="3293520">And</span> <span m="3295350">we</span> <span m="3295470">can</span> <span m="3295620">see</span> <span m="3296010">that</span> <span m="3298720">this</span> <span m="3298940">particular</span> <span m="3301120">sequence</span> <span m="3301780">also</span> <span m="3302150">maps</span> <span m="3302520">to</span> <span m="3302630">the</span> <span m="3302890">same.</span> <span m="3303320">It</span> <span m="3303450">just</span> <span m="3303640">swap</span> <span m="3306040">the</span> <span m="3306150">positions.</span> <span m="3306840">So</span> <span m="3307160">we</span> <span m="3307270">have</span> <span m="3307560">r2,</span> <span m="3308240">c2,</span> <span m="3308830">r1,</span> <span m="3309520">and</span> <span m="3309700">c1</span> <span m="3310680">also</span> <span m="3311090">maps</span> <span m="3311550">under</span> <span m="3311820">f</span> <span m="3312530">to</span> <span m="3312650">the</span> <span m="3312800">same</span> <span m="3313510">configuration.</span> </p>
<p><span m="3314770">And</span> <span m="3316830">those</span> <span m="3316980">are</span> <span m="3317090">the</span> <span m="3317210">exact</span> <span m="3317780">2</span> <span m="3318150">possibilities</span> <span m="3318960">that</span> <span m="3319100">can</span> <span m="3319240">happen</span> <span m="3321540">that</span> <span m="3322090">map</span> <span m="3322350">to</span> <span m="3322460">this</span> <span m="3322640">configuration.</span> <span m="3324230">Every</span> <span m="3324450">valid</span> <span m="3324910">configuration</span> <span m="3325820">is</span> <span m="3326050">mapped</span> <span m="3326310">to</span> <span m="3326570">exactly</span> <span m="3326860">2</span> <span m="3327150">times.</span> <span m="3328290">So</span> <span m="3328460">now</span> <span m="3328610">we</span> <span m="3328730">can</span> <span m="3328880">use</span> <span m="3329070">the</span> <span m="3329130">division</span> <span m="3329650">rule.</span> <span m="3330780">Because</span> <span m="3331150">f</span> <span m="3331350">is</span> <span m="3331570">2</span> <span m="3331760">to</span> <span m="3331960">1.</span> </p>
<p><span m="3332570">So</span> <span m="3333960">f</span> <span m="3335130">is</span> <span m="3335530">2</span> <span m="3337400">to</span> <span m="3337650">1.</span> <span m="3339240">What</span> <span m="3339350">does</span> <span m="3339560">that</span> <span m="3339680">mean</span> <span m="3340010">if</span> <span m="3340150">you</span> <span m="3340260">apply the</span> <span m="3340560">division</span> <span m="3341030">rule?</span> <span m="3341720">It</span> <span m="3341880">means</span> <span m="3342180">that</span> <span m="3343440">the</span> <span m="3343580">cardinality</span> <span m="3344750">of</span> <span m="3346160">all those</span> <span m="3346510">sequences</span> <span m="3347220">is</span> <span m="3347350">equal</span> <span m="3347620">to</span> <span m="3347760">2</span> <span m="3348100">times</span> <span m="3348510">the</span> <span m="3348590">cardinality</span> <span m="3349500">of</span> <span m="3350400">valid</span> <span m="3350900">configurations.</span> <span m="3352420">Or</span> <span m="3352580">in</span> <span m="3352790">other</span> <span m="3352870">words,</span> <span m="3355820">the</span> <span m="3355920">cardinality</span> <span m="3356480">for</span> <span m="3356580">all</span> <span m="3356760">the</span> <span m="3356830">valid</span> <span m="3357180">configurations</span> <span m="3358070">is</span> <span m="3358250">the</span> <span m="3358330">cardinality</span> <span m="3359140">of</span> <span m="3359340">all</span> <span m="3359720">those</span> <span m="3360020">possible</span> <span m="3360420">sequences</span> <span m="3360850">divided</span> <span m="3361480">by</span> <span m="3361620">2.</span> </p>
<p><span m="3363840">So</span> <span m="3364000">now</span> <span m="3364200">we</span> <span m="3364320">can</span> <span m="3364460">start</span> <span m="3364800">counting</span> <span m="3365820">x</span> <span m="3366050">over</span> <span m="3366400">here.</span> <span m="3367720">So</span> <span m="3367780">how</span> <span m="3367900">do</span> <span m="3368000">we</span> <span m="3368100">do</span> <span m="3368210">that?</span> <span m="3368890">Well, I'm going</span> <span m="3369310">to use</span> <span m="3369550">something</span> <span m="3369930">similar</span> <span m="3370520">as</span> <span m="3370700">what</span> <span m="3370870">we</span> <span m="3370990">did</span> <span m="3371240">when</span> <span m="3371380">we</span> <span m="3371670">were</span> <span m="3371820">counting</span> <span m="3372160">permutations.</span> <span m="3373640">And</span> <span m="3373890">we'll</span> <span m="3374440">generalize</span> <span m="3375080">it</span> <span m="3375180">in</span> <span m="3375310">a</span> <span m="3375350">moment.</span> </p>
<p><span m="3377410">So</span> <span m="3377610">how</span> <span m="3377730">do</span> <span m="3377860">we</span> <span m="3377980">go</span> <span m="3378190">about</span> <span m="3378530">this?</span> <span m="3378880">Well,</span> <span m="3379310">let's</span> <span m="3379550">have</span> <span m="3379740">a</span> <span m="3379800">look.</span> <span m="3380320">If</span> <span m="3380580">I</span> <span m="3380730">have</span> <span m="3382390">r1,</span> <span m="3384274">c1,</span> <span m="3385490">and</span> <span m="3386020">r2,</span> <span m="3387010">and</span> <span m="3387230">c2,</span> <span m="3389210">so</span> <span m="3389400">this</span> <span m="3389650">is</span> <span m="3389800">a</span> <span m="3389850">sequence.</span> <span m="3392410">So</span> <span m="3392520">how</span> <span m="3392680">many</span> <span m="3392950">choices</span> <span m="3393470">do</span> <span m="3393640">I</span> <span m="3393740">have?</span> </p>
<p><span m="3395450">Well,</span> <span m="3395650">a</span> <span m="3395700">chessboard</span> <span m="3396230">has</span> <span m="3396620">8</span> <span m="3396820">rows.</span> <span m="3398130">So</span> <span m="3398330">I</span> <span m="3398410">can</span> <span m="3398630">choose</span> <span m="3400470">8</span> <span m="3401460">possible</span> <span m="3402620">rows</span> <span m="3403110">for</span> <span m="3403600">the</span> <span m="3403690">first</span> <span m="3404040">rook.</span> <span m="3406620">It</span> <span m="3406740">also</span> <span m="3406970">has</span> <span m="3407200">8</span> <span m="3407440">columns.</span> <span m="3407860">So I have</span> <span m="3408250">8</span> <span m="3408570">possible</span> <span m="3408930">choices</span> <span m="3409490">for</span> <span m="3409740">the</span> <span m="3409890">column.</span> </p>
<p><span m="3412190">But</span> <span m="3412370">what</span> <span m="3412500">about</span> <span m="3412720">the</span> <span m="3412790">second</span> <span m="3413200">rook?</span> <span m="3414180">Well, the</span> <span m="3414480">second</span> <span m="3414820">rook</span> <span m="3416550">can</span> <span m="3416720">be</span> <span m="3416890">on</span> <span m="3417110">any</span> <span m="3417420">row</span> <span m="3417680">except</span> <span m="3418130">the</span> <span m="3418250">one</span> <span m="3418690">that I've</span> <span m="3419090">already</span> <span m="3419480">selected</span> <span m="3420080">for</span> <span m="3420230">the</span> <span m="3420310">first.</span> <span m="3421590">So</span> <span m="3421940">this</span> <span m="3422230">8</span> <span m="3422480">minus</span> <span m="3422900">1,</span> <span m="3423220">we</span> <span m="3423340">have</span> <span m="3423500">7</span> <span m="3424110">possible</span> <span m="3424520">choices</span> <span m="3426590">to</span> <span m="3426730">select</span> <span m="3427790">the</span> <span m="3427910">row</span> <span m="3428460">for</span> <span m="3428790">the</span> <span m="3428910">second</span> <span m="3429260">rook.</span> </p>
<p><span m="3429530">It</span> <span m="3429650">must</span> <span m="3429880">be</span> <span m="3429980">different</span> <span m="3430430">from</span> <span m="3430590">the</span> <span m="3430700">one</span> <span m="3431410">that</span> <span m="3431560">was</span> <span m="3431720">already</span> <span m="3432040">selected.</span> <span m="3432600">And</span> <span m="3432760">I have</span> <span m="3432830">7</span> <span m="3433180">possible</span> <span m="3433540">choices.</span> <span m="3434620">And</span> <span m="3434770">similarly</span> <span m="3435310">for</span> <span m="3436570">this</span> <span m="3436830">particular</span> <span m="3437320">column</span> <span m="3437650">as well,</span> <span m="3437930">the</span> <span m="3438040">column</span> <span m="3438340">has</span> <span m="3438510">to</span> <span m="3438600">be</span> <span m="3438720">different.</span> <span m="3439990">So</span> <span m="3440150">how</span> <span m="3440310">many</span> <span m="3440610">choices</span> <span m="3441050">do</span> <span m="3441220">I</span> <span m="3441320">have?</span> </p>
<p><span m="3442080">Well,</span> <span m="3442530">it's</span> <span m="3442740">not</span> <span m="3443020">8.</span> <span m="3443310">It's</span> <span m="3443630">one</span> <span m="3443830">less,</span> <span m="3444160">because</span> <span m="3444490">I've</span> <span m="3444600">already</span> <span m="3444920">selected</span> <span m="3445440">the</span> <span m="3445530">one</span> <span m="3445750">for</span> <span m="3446350">the</span> <span m="3446460">first</span> <span m="3447110">rook.</span> <span m="3447830">So</span> <span m="3448000">I</span> <span m="3448100">have</span> <span m="3448280">7</span> <span m="3448710">choices.</span> <span m="3451100">So</span> <span m="3452870">the</span> <span m="3453000">cardinality</span> <span m="3453620">of</span> <span m="3453760">x</span> <span m="3454050">is</span> <span m="3454190">actually</span> <span m="3454530">equal</span> <span m="3455500">to</span> <span m="3455650">8</span> <span m="3455950">times</span> <span m="3456200">8</span> <span m="3456450">times</span> <span m="3456670">7</span> <span m="3456980">times</span> <span m="3457220">7.</span> </p>
<p><span m="3458220">So</span> <span m="3458430">it's</span> <span m="3458650">8</span> <span m="3459210">times</span> <span m="3459460">7</span> <span m="3459830">squared</span> <span m="3460930">divided</span> <span m="3461400">by</span> <span m="3461550">2.</span> <span m="3463520">So</span> <span m="3463690">now</span> <span m="3463840">we</span> <span m="3463940">have</span> <span m="3464090">to</span> <span m="3464130">counted,</span> <span m="3464900">by</span> <span m="3465030">using</span> <span m="3465360">the division</span> <span m="3465920">rule,</span> <span m="3466670">we</span> <span m="3467030">have</span> <span m="3467290">to</span> <span m="3467380">the</span> <span m="3467480">divide</span> <span m="3467820">this by</span> <span m="3468340">2.</span> <span m="3469270">We have</span> <span m="3469480">counted</span> <span m="3470070">the</span> <span m="3470260">total</span> <span m="3470510">number</span> <span m="3470770">of</span> <span m="3470840">valid</span> <span m="3471140">configurations.</span> </p>
<p><span m="3475460">So</span> <span m="3475640">now</span> <span m="3476200">we are</span> <span m="3476410">going</span> <span m="3476580">to</span> <span m="3476680">generalize</span> <span m="3477330">this</span> <span m="3477900">principle</span> <span m="3478560">that</span> <span m="3478760">we have</span> <span m="3478940">talked</span> <span m="3479260">about</span> <span m="3479570">here.</span> <span m="3480580">And</span> <span m="3483260">we</span> <span m="3483580">will</span> <span m="3483800">do</span> <span m="3483960">that</span> <span m="3484190">over</span> <span m="3484570">here</span> <span m="3484850">I</span> <span m="3484920">think.</span> <span m="3485980">Yup.</span> <span m="3487870">And</span> <span m="3488730">that's</span> <span m="3489370">the</span> <span m="3489990">generalized</span> <span m="3490690">product</span> <span m="3491030">rule.</span> </p>
<p><span m="3495210">So</span> <span m="3495390">the</span> <span m="3495470">generalized</span> <span m="3495970">product</span> <span m="3496300">rule</span> <span m="3499110">is</span> <span m="3499330">as</span> <span m="3499480">follows.</span> <span m="3501460">It's</span> <span m="3501690">essentially</span> <span m="3503590">saying</span> <span m="3504220">that</span> <span m="3504490">if</span> <span m="3504780">we</span> <span m="3505200">have</span> <span m="3505380">a</span> <span m="3505480">set</span> <span m="3507850">of</span> <span m="3508020">sequences</span> <span m="3508610">of</span> <span m="3508720">length</span> <span m="3509050">k,</span> <span m="3510480">then</span> <span m="3510950">how</span> <span m="3511130">can</span> <span m="3511280">we</span> <span m="3511400">count</span> <span m="3511710">those?</span> <span m="3512320">Well,</span> <span m="3513560">if</span> <span m="3513750">we</span> <span m="3513840">know</span> <span m="3514380">the</span> <span m="3514480">following</span> <span m="3515360">properties--</span> <span m="3515950">well, let</span> <span m="3516160">me</span> <span m="3517020">first</span> <span m="3517440">write</span> <span m="3517670">out</span> <span m="3520420">the</span> <span m="3520520">generalized</span> <span m="3525007">product</span> <span m="3525486">rule</span> <span m="3530760">is</span> <span m="3530960">as</span> <span m="3531100">follows.</span> </p>
<p><span m="3533050">Let</span> <span m="3533290">S</span> <span m="3535830">be a</span> <span m="3536130">set</span> <span m="3538450">of</span> <span m="3538900">length</span> <span m="3541970">k</span> <span m="3542890">sequences.</span> <span m="3547300">Then</span> <span m="3547550">I</span> <span m="3547630">know</span> <span m="3547935">that</span> <span m="3548240">if</span> <span m="3552960">there</span> <span m="3553240">are</span> <span m="3557370">n1</span> <span m="3558530">possible</span> <span m="3559040">first</span> <span m="3559370">entries</span> <span m="3566940">and</span> <span m="3567160">if</span> <span m="3567290">I</span> <span m="3567380">know</span> <span m="3567650">that</span> <span m="3568260">once</span> <span m="3568530">I've</span> <span m="3568670">selected</span> <span m="3569200">my</span> <span m="3569340">first</span> <span m="3569670">entry,</span> <span m="3570970">there</span> <span m="3571230">are</span> <span m="3571490">n2</span> <span m="3572100">possible</span> <span m="3573050">second</span> <span m="3573480">entries</span> <span m="3579090">for</span> <span m="3579460">each</span> <span m="3582630">first</span> <span m="3583070">entry.</span> <span m="3595020">And</span> <span m="3595390">if</span> <span m="3595570">I</span> <span m="3595620">continue</span> <span m="3596050">like</span> <span m="3596280">this,</span> <span m="3597110">my</span> <span m="3597240">choice</span> <span m="3597640">for</span> <span m="3597730">the</span> <span m="3597840">third</span> <span m="3598720">term</span> <span m="3599120">in the</span> <span m="3599320">sequence</span> <span m="3600710">is</span> <span m="3604750">I've</span> <span m="3604960">always</span> <span m="3605330">n3</span> <span m="3606210">possible</span> <span m="3606680">choices</span> <span m="3607930">given</span> <span m="3608540">my</span> <span m="3608710">selection</span> <span m="3609200">for</span> <span m="3609360">the</span> <span m="3609440">first</span> <span m="3609800">2</span> <span m="3610110">entries.</span> </p>
<p><span m="3611990">So</span> <span m="3612120">if</span> <span m="3612330">I</span> <span m="3612790">have</span> <span m="3613030">that</span> <span m="3613250">property</span> <span m="3613720">that</span> <span m="3613890">continues</span> <span m="3614440">in</span> <span m="3614580">that way,</span> <span m="3614950">so</span> <span m="3615110">let</span> <span m="3615240">me</span> <span m="3615330">write</span> <span m="3615520">it</span> <span m="3615640">out.</span> <span m="3616080">So</span> <span m="3616330">we</span> <span m="3616440">have</span> <span m="3616680">n3</span> <span m="3618940">possible</span> <span m="3622800">third</span> <span m="3624520">entries</span> <span m="3626458">for</span> <span m="3626924">each</span> <span m="3630660">combination</span> <span m="3631490">in this</span> <span m="3631988">case</span> <span m="3635980">of</span> <span m="3637660">first</span> <span m="3639210">together</span> <span m="3639930">with</span> <span m="3640090">second</span> <span m="3642740">entries.</span> <span m="3645370">And</span> <span m="3645770">if</span> <span m="3646000">I</span> <span m="3646050">continue</span> <span m="3646570">this</span> <span m="3647000">all</span> <span m="3647270">the</span> <span m="3647370">way</span> <span m="3648140">to</span> <span m="3648320">nk,</span> <span m="3649890">so</span> <span m="3650260">nk</span> <span m="3650580">possible</span> <span m="3651900">kth</span> <span m="3652230">entries</span> <span m="3653010">for</span> <span m="3653280">each</span> <span m="3653550">combination</span> <span m="3654320">of</span> <span m="3654600">all</span> <span m="3654810">the</span> <span m="3654910">previous</span> <span m="3655330">entries,</span> <span m="3660100">then</span> <span m="3660700">I</span> <span m="3660850">know</span> <span m="3662070">that</span> <span m="3662460">the</span> <span m="3662570">set</span> <span m="3663480">S</span> <span m="3664100">can</span> <span m="3664250">be</span> <span m="3664390">counted</span> <span m="3664730">as,</span> <span m="3666430">well,</span> <span m="3666780">I've</span> <span m="3667100">n1</span> <span m="3667730">possible</span> <span m="3668180">choices</span> <span m="3668540">for</span> <span m="3668650">the</span> <span m="3668740">first</span> <span m="3669100">entry.</span> </p>
<p><span m="3669560">Once</span> <span m="3669680">I've</span> <span m="3669780">chosen</span> <span m="3670210">to fix</span> <span m="3670390">that</span> <span m="3670760">one,</span> <span m="3671350">I have</span> <span m="3671580">n2</span> <span m="3671930">possible</span> <span m="3672300">choices</span> <span m="3672660">for the</span> <span m="3672900">second,</span> <span m="3674770">then</span> <span m="3675010">n3</span> <span m="3675500">possible</span> <span m="3675910">choice</span> <span m="3676250">for</span> <span m="3676370">the</span> <span m="3676610">third.</span> <span m="3676830">And</span> <span m="3677190">I</span> <span m="3677330">go</span> <span m="3677510">all</span> <span m="3677740">the</span> <span m="3677830">way</span> <span m="3678030">to</span> <span m="3678760">nk.</span> <span m="3681520">Well, let's</span> <span m="3681930">first</span> <span m="3682150">talk</span> <span m="3682340">about</span> <span m="3682590">it</span> <span m="3682730">from</span> <span m="3682940">the</span> <span m="3683030">perspective</span> <span m="3683780">of</span> <span m="3683930">the</span> <span m="3684010">chess</span> <span m="3684350">problem</span> <span m="3684760">here.</span> </p>
<p><span m="3685980">I</span> <span m="3686120">got</span> <span m="3686780">8</span> <span m="3687120">possible</span> <span m="3687550">choices</span> <span m="3688020">for</span> <span m="3688230">r1.</span> <span m="3689960">Given</span> <span m="3690320">r1,</span> <span m="3691420">I</span> <span m="3691520">don't</span> <span m="3691910">care.</span> <span m="3692130">I</span> <span m="3692170">still</span> <span m="3692490">have</span> <span m="3692720">8</span> <span m="3692970">possible</span> <span m="3693290">choices</span> <span m="3693760">for</span> <span m="3693990">the</span> <span m="3694550">column</span> <span m="3694870">here.</span> <span m="3695290">So</span> <span m="3695480">I have</span> <span m="3695610">8</span> <span m="3695770">choices</span> <span m="3696210">here.</span> <span m="3697270">But</span> <span m="3697540">for the</span> <span m="3697610">third</span> <span m="3697980">one,</span> <span m="3698290">once</span> <span m="3698560">I</span> <span m="3698720">have</span> <span m="3700180">selected</span> <span m="3700760">r1 and</span> <span m="3701246">c1,</span> <span m="3702220">I</span> <span m="3702430">only</span> <span m="3702780">have</span> <span m="3702980">7</span> <span m="3703460">choices</span> <span m="3703940">left</span> <span m="3704260">for</span> <span m="3704370">r2</span> <span m="3705980">and</span> <span m="3706210">7</span> <span m="3706560">choices</span> <span m="3706980">left</span> <span m="3707190">for</span> <span m="3707330">c2.</span> </p>
<p><span m="3708160">So</span> <span m="3708310">that's</span> <span m="3708540">an</span> <span m="3708630">example</span> <span m="3709300">where</span> <span m="3710140">we</span> <span m="3710250">use</span> <span m="3710800">this</span> <span m="3711110">particular</span> <span m="3712490">generalized</span> <span m="3713030">product</span> <span m="3713380">rule.</span> <span m="3714160">Also</span> <span m="3715920">when</span> <span m="3716150">we</span> <span m="3716290">were</span> <span m="3716440">counting</span> <span m="3717000">the</span> <span m="3717110">number</span> <span m="3717280">of</span> <span m="3717470">permutations,</span> <span m="3719210">we</span> <span m="3719340">were</span> <span m="3719490">saying</span> <span m="3720130">we</span> <span m="3720250">can</span> <span m="3720440">fix</span> <span m="3720820">the</span> <span m="3720930">first</span> <span m="3721970">entry</span> <span m="3722390">of</span> <span m="3722500">a</span> <span m="3722630">permutation</span> <span m="3723440">in</span> <span m="3723730">n ways</span> <span m="3724300">if I have</span> <span m="3724740">a</span> <span m="3724880">permutation</span> <span m="3725510">for</span> <span m="3725700">n</span> <span m="3725900">elements.</span> <span m="3727160">And</span> <span m="3727280">then</span> <span m="3727440">I</span> <span m="3727550">have</span> <span m="3728680">the</span> <span m="3728910">second</span> <span m="3732130">entry,</span> <span m="3732710">the</span> <span m="3732800">second</span> <span m="3733150">term,</span> <span m="3733640">of</span> <span m="3733760">a</span> <span m="3733830">permutation</span> <span m="3734580">has</span> <span m="3734880">only</span> <span m="3735200">n</span> <span m="3735350">minus</span> <span m="3735660">1</span> <span m="3735880">choices.</span> <span m="3736410">Because</span> <span m="3736780">I've</span> <span m="3736900">already</span> <span m="3737280">chosen</span> <span m="3737720">one.</span> </p>
<p><span m="3738770">And</span> <span m="3738910">the</span> <span m="3738960">next</span> <span m="3739270">one</span> <span m="3739410">has</span> <span m="3739550">n</span> <span m="3739680">minus</span> <span m="3740030">3</span> <span m="3740240">choices.</span> <span m="3740850">Because</span> <span m="3741210">I've</span> <span m="3741440">already</span> <span m="3741740">selected</span> <span m="3742390">2</span> <span m="3742640">of</span> <span m="3742800">them.</span> <span m="3743020">So</span> <span m="3743200">I have</span> <span m="3743390">only</span> <span m="3743530">n</span> <span m="3743660">minus</span> <span m="3743950">2</span> <span m="3744070">choices</span> <span m="3744460">left.</span> <span m="3745350">Then</span> <span m="3745520">I</span> <span m="3745610">have</span> <span m="3745770">n</span> <span m="3745910">minus</span> <span m="3746220">3</span> <span m="3746430">choices.</span> <span m="3746830">Because</span> <span m="3747120">I've</span> <span m="3747230">already</span> <span m="3747570">selected</span> <span m="3748000">3</span> <span m="3748400">and</span> <span m="3748580">so</span> <span m="3748810">on.</span> </p>
<p><span m="3749430">And</span> <span m="3749560">I</span> <span m="3749600">get</span> <span m="3749830">n</span> <span m="3750000">factorial.</span> <span m="3751870">So</span> <span m="3751900">that's</span> <span m="3752090">the</span> <span m="3752180">same</span> <span m="3752440">kind</span> <span m="3752710">of</span> <span m="3753250">principle</span> <span m="3753890">that</span> <span m="3754040">we have</span> <span m="3754290">here.</span> <span m="3756070">So</span> <span m="3756340">let</span> <span m="3756500">me</span> <span m="3756900">give</span> <span m="3757180">an</span> <span m="3757280">example</span> <span m="3759230">where we</span> <span m="3759320">can</span> <span m="3759480">see</span> <span m="3761300">how</span> <span m="3761470">this</span> <span m="3761680">works.</span> </p>
<p><span m="3777550">So</span> <span m="3777680">what</span> <span m="3777780">do</span> <span m="3777860">we</span> <span m="3777980">do?</span> <span m="3779700">In</span> <span m="3779820">this</span> <span m="3779990">example,</span> <span m="3780870">we</span> <span m="3780970">want</span> <span m="3781220">to</span> <span m="3782010">count</span> <span m="3782410">the</span> <span m="3782490">number</span> <span m="3782800">of</span> <span m="3782950">committees.</span> <span m="3783920">So</span> <span m="3784120">it's</span> <span m="3784445">the exact</span> <span m="3784770">same</span> <span m="3785040">kind</span> <span m="3785270">of</span> <span m="3785360">principle</span> <span m="3785635">that</span> <span m="3785910">we are</span> <span m="3786350">going to</span> <span m="3786450">talk</span> <span m="3786750">about.</span> </p>
<p><span m="3788170">So</span> <span m="3788390">we are</span> <span m="3788500">going</span> <span m="3788760">to</span> <span m="3789890">count</span> <span m="3790150">the</span> <span m="3790230">number</span> <span m="3790480">of</span> <span m="3790610">communities</span> <span m="3792870">described</span> <span m="3793390">by</span> <span m="3793560">sequence</span> <span m="3794070">x,</span> <span m="3794486">y,</span> <span m="3794902">z,</span> <span m="3795810">where</span> <span m="3796080">x</span> <span m="3796460">the</span> <span m="3796550">first</span> <span m="3797010">one</span> <span m="3797380">is,</span> <span m="3797580">say,</span> <span m="3798370">the</span> <span m="3798610">leader</span> <span m="3799010">of</span> <span m="3799180">the</span> <span m="3799270">committee.</span> <span m="3801560">The</span> <span m="3801830">second</span> <span m="3802200">one</span> <span m="3802790">indicates</span> <span m="3803490">the</span> <span m="3803570">secretary.</span> <span m="3805790">The</span> <span m="3805930">third</span> <span m="3806350">one is</span> <span m="3806720">some</span> <span m="3807220">consultant.</span> <span m="3809200">So</span> <span m="3809380">they're</span> <span m="3809490">all</span> <span m="3809700">different.</span> <span m="3810880">They have all</span> <span m="3811170">different</span> <span m="3811530">roles.</span> </p>
<p><span m="3813070">And</span> <span m="3813500">such</span> <span m="3813770">a</span> <span m="3813840">committee</span> <span m="3814330">is</span> <span m="3814820">elected</span> <span m="3816860">from</span> <span m="3820600">n</span> <span m="3820920">members.</span> <span m="3824260">And</span> <span m="3824515">in</span> <span m="3824770">how</span> <span m="3825040">many</span> <span m="3825290">ways</span> <span m="3825570">can</span> <span m="3825720">I</span> <span m="3825790">do</span> <span m="3825980">this?</span> <span m="3826690">Well,</span> <span m="3827470">I</span> <span m="3827650">have</span> <span m="3828100">n ways</span> <span m="3828630">to</span> <span m="3828760">choose</span> <span m="3829130">my</span> <span m="3829290">first</span> <span m="3832160">term</span> <span m="3832480">in</span> <span m="3832550">the</span> <span m="3832640">sequence.</span> <span m="3833290">I have</span> <span m="3833460">n</span> <span m="3833680">ways</span> <span m="3833960">to</span> <span m="3834080">choose</span> <span m="3834430">the</span> <span m="3834450">leader.</span> </p>
<p><span m="3835590">So</span> <span m="3835770">there's</span> <span m="3836040">n ways</span> <span m="3838273">to</span> <span m="3838754">choose x.</span> <span m="3843090">How</span> <span m="3843220">many</span> <span m="3843450">ways</span> <span m="3843700">do I have</span> <span m="3844050">to</span> <span m="3844150">choose</span> <span m="3844450">y?</span> <span m="3844680">Well,</span> <span m="3844910">if</span> <span m="3845070">I've</span> <span m="3845210">chosen</span> <span m="3845580">already a</span> <span m="3845940">leader,</span> <span m="3846880">I</span> <span m="3846940">need</span> <span m="3847110">to</span> <span m="3847220">choose</span> <span m="3847480">someone</span> <span m="3847840">else.</span> <span m="3848400">So</span> <span m="3848610">I</span> <span m="3848710">have</span> <span m="3848960">n</span> <span m="3849180">minus</span> <span m="3849610">1</span> <span m="3849810">members</span> <span m="3850260">left,</span> <span m="3852170">n</span> <span m="3852340">minus</span> <span m="3852610">1</span> <span m="3852710">ways</span> <span m="3853140">to</span> <span m="3853270">choose</span> <span m="3854150">y.</span> </p>
<p><span m="3857760">I'm</span> <span m="3857910">just</span> <span m="3858150">not</span> <span m="3858310">allowed</span> <span m="3859470">to</span> <span m="3859860">choose</span> <span m="3860180">x.</span> <span m="3863690">And</span> <span m="3863850">then</span> <span m="3864670">I</span> <span m="3864800">will</span> <span m="3864930">have</span> <span m="3865200">n</span> <span m="3865360">minus</span> <span m="3865610">2</span> <span m="3866130">ways</span> <span m="3867000">to</span> <span m="3867120">choose</span> <span m="3867550">a</span> <span m="3867640">z</span> <span m="3869840">except</span> <span m="3871850">x</span> <span m="3872460">and</span> <span m="3872720">y.</span> <span m="3874100">And so</span> <span m="3874270">for</span> <span m="3874550">each</span> <span m="3876060">x,</span> <span m="3876570">I have</span> <span m="3876760">only</span> <span m="3877010">n</span> <span m="3877180">minus</span> <span m="3877580">1</span> <span m="3877840">ways</span> <span m="3878230">to</span> <span m="3878340">choose</span> <span m="3878730">y.</span> <span m="3879270">For</span> <span m="3879460">each</span> <span m="3879850">x</span> <span m="3880140">and</span> <span m="3880370">y,</span> <span m="3881070">I have</span> <span m="3881280">only</span> <span m="3881450">n</span> <span m="3881600">minus</span> <span m="3881960">2</span> <span m="3882150">ways</span> <span m="3882520">to choose</span> <span m="3882740">z.</span> </p>
<p><span m="3884010">So</span> <span m="3884150">if</span> <span m="3884320">I</span> <span m="3884350">multiply</span> <span m="3884900">all</span> <span m="3885020">this</span> <span m="3885250">together,</span> <span m="3886640">I</span> <span m="3886750">get</span> <span m="3887050">n</span> <span m="3887440">times</span> <span m="3887840">n</span> <span m="3888350">minus</span> <span m="3888730">1</span> <span m="3889460">times</span> <span m="3889790">n</span> <span m="3889980">minus</span> <span m="3890400">2</span> <span m="3891190">to</span> <span m="3891430">choose</span> <span m="3892770">all</span> <span m="3892960">these</span> <span m="3893140">committee--</span> <span m="3894110">this</span> <span m="3894620">is</span> <span m="3894710">the</span> <span m="3894800">total</span> <span m="3895080">number</span> <span m="3895400">of</span> <span m="3895520">possible</span> <span m="3895930">committees</span> <span m="3896360">that</span> <span m="3896590">I</span> <span m="3896640">can</span> <span m="3896830">select</span> <span m="3897620">from</span> <span m="3897920">an</span> <span m="3898380">n</span> <span m="3898730">member</span> <span m="3899800">set</span> <span m="3900040">of</span> <span m="3900160">people.</span> <span m="3901960">So</span> <span m="3902020">let's</span> <span m="3902260">go</span> <span m="3902430">to</span> <span m="3903510">a</span> <span m="3903540">little</span> <span m="3903770">bit</span> <span m="3903940">of</span> <span m="3904060">a</span> <span m="3904120">different</span> <span m="3905650">example</span> <span m="3906370">that</span> <span m="3906450">uses</span> <span m="3906810">the</span> <span m="3906880">same</span> <span m="3907130">principle.</span> <span m="3910660">OK.</span> <span m="3910930">Let's</span> <span m="3913080">make</span> <span m="3913500">some</span> <span m="3913670">space.</span> </p>
<p><span m="3918570">In</span> <span m="3918700">the</span> <span m="3918800">second</span> <span m="3919220">problem,</span> <span m="3921900">I will</span> <span m="3922160">define</span> <span m="3922560">to you</span> <span m="3922790">a</span> <span m="3922950">defective</span> <span m="3923920">dollar</span> <span m="3924240">bill.</span> <span m="3925170">It's</span> <span m="3925270">not really</span> <span m="3925640">defective.</span> <span m="3926190">But</span> <span m="3927170">it's</span> <span m="3927340">a</span> <span m="3927410">property</span> <span m="3927880">that we</span> <span m="3928050">will</span> <span m="3928280">assign</span> <span m="3928710">to</span> <span m="3928830">dollar</span> <span m="3929170">bill.</span> </p>
<p><span m="3930710">And</span> <span m="3930830">you</span> <span m="3930920">can</span> <span m="3931070">check</span> <span m="3931340">for</span> <span m="3931490">yourself</span> <span m="3931990">whether</span> <span m="3932020">you have</span> <span m="3932320">one</span> <span m="3932600">in your</span> <span m="3932730">wallet.</span> <span m="3934410">So</span> <span m="3935080">let's</span> <span m="3935300">define</span> <span m="3935750">a defective</span> <span m="3937450">dollar</span> <span m="3937910">bill</span> <span m="3938230">to</span> <span m="3938370">have</span> <span m="3939210">the</span> <span m="3939400">property</span> <span m="3941190">that</span> <span m="3941360">if</span> <span m="3941500">you</span> <span m="3941650">look</span> <span m="3941860">at</span> <span m="3942160">the</span> <span m="3942990">8-bit</span> <span m="3947640">serial</span> <span m="3948050">number,</span> <span m="3952590">some</span> <span m="3952890">of</span> <span m="3952980">the</span> <span m="3953090">digits</span> <span m="3953540">appear</span> <span m="3954040">more</span> <span m="3954620">than</span> <span m="3954810">once.</span> <span m="3955590">So</span> <span m="3956440">some</span> <span m="3956970">digit</span> <span m="3960790">appears</span> <span m="3963570">more</span> <span m="3967000">than</span> <span m="3967330">once</span> <span m="3971560">in</span> <span m="3972690">the</span> <span m="3972800">8-bit</span> <span m="3973760">serial</span> <span m="3974230">number.</span> </p>
<p><span m="3979430">So</span> <span m="3979530">you</span> <span m="3979690">can</span> <span m="3979860">check</span> <span m="3980110">our</span> <span m="3980230">own</span> <span m="3980360">wallet</span> <span m="3980680">and</span> <span m="3980790">check</span> <span m="3981020">for</span> <span m="3981240">your</span> <span m="3981400">$1</span> <span m="3981830">bills</span> <span m="3982930">and</span> <span m="3983110">check</span> <span m="3983330">whether</span> <span m="3983700">you have</span> <span m="3983770">a</span> <span m="3983850">defective</span> <span m="3984160">dollar.</span> <span m="3984490">This</span> <span m="3984600">seems</span> <span m="3984840">to</span> <span m="3984900">be a</span> <span m="3985100">pretty</span> <span m="3985860">specific</span> <span m="3986530">and</span> <span m="3986620">rare</span> <span m="3986980">property,</span> <span m="3987480">right?</span> <span m="3988700">Well,</span> <span m="3992250">check you</span> <span m="3992540">dollar</span> <span m="3992800">bills.</span> </p>
<p><span m="3993560">You'll</span> <span m="3993720">figure</span> <span m="3993870">out</span> <span m="3994210">that</span> <span m="3994300">you</span> <span m="3994440">have</span> <span m="3995200">probably</span> <span m="3995680">a</span> <span m="3995760">defective</span> <span m="3996120">dollar</span> <span m="3996370">bill</span> <span m="3996610">in your</span> <span m="3996770">wallet.</span> <span m="3997130">So</span> <span m="3997260">that's</span> <span m="3997430">kind</span> <span m="3997630">of</span> <span m="3997740">weird.</span> <span m="3998580">But</span> <span m="3998740">it</span> <span m="3998830">seems</span> <span m="3999790">this</span> <span m="3999990">property.</span> <span m="4001120">If</span> <span m="4001180">you</span> <span m="4001290">look</span> <span m="4001470">at</span> <span m="4001550">that,</span> <span m="4001830">it</span> <span m="4001960">seems</span> <span m="4002220">to</span> <span m="4002330">be</span> <span m="4003220">something</span> <span m="4003600">that</span> <span m="4003740">is</span> <span m="4003920">maybe a</span> <span m="4004090">little bit</span> <span m="4004420">more</span> <span m="4004620">common</span> <span m="4004900">than</span> <span m="4005180">we thought it</span> <span m="4005750">is.</span> </p>
<p><span m="4006580">It</span> <span m="4007000">seems</span> <span m="4007330">to</span> <span m="4007400">be</span> <span m="4007530">so</span> <span m="4007710">special.</span> <span m="4008590">So</span> <span m="4008620">let's do</span> <span m="4008750">a</span> <span m="4009000">counting</span> <span m="4009410">argument</span> <span m="4010290">and</span> <span m="4010420">find</span> <span m="4010710">out</span> <span m="4010850">what's</span> <span m="4011030">happening</span> <span m="4011380">here.</span> <span m="4012590">So</span> <span m="4012730">let's</span> <span m="4014170">look</span> <span m="4014400">at</span> <span m="4014480">a</span> <span m="4014530">fraction</span> <span m="4016680">of</span> <span m="4018310">the</span> <span m="4018430">non-defective.</span> <span m="4019510">So</span> <span m="4019730">we are</span> <span m="4019890">counting</span> <span m="4020180">the</span> <span m="4020310">opposite,</span> <span m="4024750">the</span> <span m="4025090">non-defective</span> <span m="4026210">bills.</span> </p>
<p><span m="4027540">Well,</span> <span m="4027740">that's</span> <span m="4027960">the</span> <span m="4028040">number</span> <span m="4029040">of</span> <span m="4029410">non-defective</span> <span m="4031950">serial</span> <span m="4032380">numbers</span> <span m="4041090">divided</span> <span m="4041610">by</span> <span m="4042540">the</span> <span m="4042700">total</span> <span m="4045060">number</span> <span m="4045820">of</span> <span m="4046500">serial</span> <span m="4046900">numbers.</span> <span m="4051680">And</span> <span m="4052400">let's</span> <span m="4052670">call</span> <span m="4052950">these</span> <span m="4053740">small</span> <span m="4054260">x</span> <span m="4055200">and</span> <span m="4055810">y</span> <span m="4057240">and</span> <span m="4057820">count</span> <span m="4058195">these.</span> </p>
<p><span m="4060030">So</span> <span m="4060190">let's</span> <span m="4060420">see.</span> <span m="4064340">So</span> <span m="4064490">first</span> <span m="4064740">of</span> <span m="4064810">all,</span> <span m="4065020">let's count</span> <span m="4065460">y.</span> <span m="4066250">Well,</span> <span m="4066360">that's</span> <span m="4066570">easy,</span> <span m="4067380">I have</span> <span m="4067770">8 digits</span> <span m="4068320">in</span> <span m="4068430">my</span> <span m="4068540">serial</span> <span m="4069000">number.</span> <span m="4069480">So</span> <span m="4069680">I have</span> <span m="4069760">10</span> <span m="4070050">choices</span> <span m="4070480">for</span> <span m="4070580">the</span> <span m="4070660">first</span> <span m="4071020">digit,</span> <span m="4072290">10</span> <span m="4072510">choices</span> <span m="4072820">for</span> <span m="4072920">the</span> <span m="4072990">second</span> <span m="4073380">one,</span> <span m="4073730">and</span> <span m="4073820">so</span> <span m="4074030">on.</span> <span m="4075170">In total,</span> <span m="4075600">I have</span> <span m="4075960">10</span> <span m="4076180">times</span> <span m="4076470">10</span> <span m="4076690">times</span> <span m="4076970">10</span> <span m="4077210">to</span> <span m="4077480">the</span> <span m="4077580">power</span> <span m="4077910">8</span> <span m="4078090">choices.</span> </p>
<p><span m="4079580">What</span> <span m="4079730">about</span> <span m="4080030">x?</span> <span m="4081530">Well,</span> <span m="4082510">I'm</span> <span m="4082690">using,</span> <span m="4083970">again,</span> <span m="4085410">our</span> <span m="4085870">generalized</span> <span m="4087660">product</span> <span m="4088040">rule</span> <span m="4088320">over</span> <span m="4088580">here.</span> <span m="4089760">Well,</span> <span m="4089990">if</span> <span m="4090170">I'm</span> <span m="4090240">going</span> <span m="4090580">to</span> <span m="4091800">have</span> <span m="4092090">a</span> <span m="4092180">non-defective</span> <span m="4093350">dollar</span> <span m="4093730">bill,</span> <span m="4094070">then</span> <span m="4094300">all</span> <span m="4094570">the</span> <span m="4094680">digits</span> <span m="4095490">in</span> <span m="4096040">the 8</span> <span m="4096160">digit</span> <span m="4097430">serial</span> <span m="4097779">number</span> <span m="4098069">have</span> <span m="4098399">to</span> <span m="4098510">be</span> <span m="4098680">different.</span> </p>
<p><span m="4100100">So</span> <span m="4100270">for</span> <span m="4100390">the</span> <span m="4100490">first</span> <span m="4100920">digit,</span> <span m="4101260">I</span> <span m="4101420">have</span> <span m="4101600">10</span> <span m="4101840">choices.</span> <span m="4103050">Now</span> <span m="4103200">that</span> <span m="4103380">I've</span> <span m="4103540">selected</span> <span m="4104060">my</span> <span m="4104200">first</span> <span m="4104540">digit,</span> <span m="4105649">I</span> <span m="4105800">have</span> <span m="4107920">9</span> <span m="4108170">digits</span> <span m="4108720">for</span> <span m="4108890">my</span> <span m="4109130">second</span> <span m="4109649">choice</span> <span m="4110639">for</span> <span m="4111050">my</span> <span m="4111180">second</span> <span m="4111819">digit</span> <span m="4112290">in the serial</span> <span m="4112740">number.</span> <span m="4114020">Then</span> <span m="4114220">I</span> <span m="4114340">have</span> <span m="4114490">8</span> <span m="4114750">possible</span> <span m="4115109">choices,</span> <span m="4115660">7,</span> <span m="4116430">because</span> <span m="4116729">I've</span> <span m="4116850">already</span> <span m="4117149">selected</span> <span m="4117649">3.</span> <span m="4117950">And I</span> <span m="4118090">cannot</span> <span m="4118359">choose</span> <span m="4118630">those</span> <span m="4118910">anymore--</span> <span m="4120090">times</span> <span m="4120380">6</span> <span m="4120939">times</span> <span m="4121810">5</span> <span m="4122120">times</span> <span m="4122420">4</span> <span m="4122779">times</span> <span m="4123220">3.</span> </p>
<p><span m="4123760">And</span> <span m="4123840">now</span> <span m="4123970">I have</span> <span m="4124460">chosen</span> <span m="4125279">an</span> <span m="4125580">8</span> <span m="4126620">digit</span> <span m="4127069">serial</span> <span m="4127439">number</span> <span m="4127859">in</span> <span m="4128040">which</span> <span m="4128330">all</span> <span m="4128600">the</span> <span m="4128680">digits</span> <span m="4129390">are</span> <span m="4129569">different.</span> <span m="4131760">OK</span> <span m="4133069">So</span> <span m="4133250">how</span> <span m="4133390">many</span> <span m="4133670">are</span> <span m="4133960">these?</span> <span m="4135680">Well,</span> <span m="4135790">this</span> <span m="4135979">is</span> <span m="4136120">actually</span> <span m="4136500">equal</span> <span m="4136800">to</span> <span m="4137010">10</span> <span m="4137500">factorial</span> <span m="4138500">divided</span> <span m="4139010">by</span> <span m="4139140">2</span> <span m="4139380">factorial.</span> <span m="4141330">And</span> <span m="4141630">it</span> <span m="4141720">turns</span> <span m="4142040">out</span> <span m="4142270">to</span> <span m="4142370">be</span> <span m="4143010">something</span> <span m="4143479">like</span> <span m="4144020">1,814,400.</span> <span m="4148100">possible</span> <span m="4148520">choices.</span> </p>
<p><span m="4150140">So</span> <span m="4150189">now</span> <span m="4150330">let's</span> <span m="4150580">look</span> <span m="4150750">at</span> <span m="4150819">the</span> <span m="4150910">fraction.</span> <span m="4152359">It</span> <span m="4152710">turns</span> <span m="4153000">out</span> <span m="4153130">if you</span> <span m="4153240">divide</span> <span m="4153700">this</span> <span m="4154310">by</span> <span m="4154500">this,</span> <span m="4155490">you</span> <span m="4155580">get a</span> <span m="4155840">really</span> <span m="4156430">very</span> <span m="4156700">small</span> <span m="4157160">fraction.</span> <span m="4158640">This</span> <span m="4158810">is</span> <span m="4158910">actually</span> <span m="4159330">equal</span> <span m="4160170">to</span> <span m="4160729">1.8144%</span> <span m="4165800">So</span> <span m="4166830">a</span> <span m="4166960">very</span> <span m="4167500">small</span> <span m="4167960">fraction</span> <span m="4168660">is</span> <span m="4168880">non-defective.</span> </p>
<p><span m="4169620">So</span> <span m="4169859">almost</span> <span m="4170330">all</span> <span m="4170660">the</span> <span m="4170740">dollars</span> <span m="4171270">are</span> <span m="4171450">sort</span> <span m="4171729">of</span> <span m="4171830">defective.</span> <span m="4172370">It</span> <span m="4172720">simply</span> <span m="4173040">means</span> <span m="4173290">that</span> <span m="4173390">they</span> <span m="4173490">have</span> <span m="4173899">this</span> <span m="4174290">special</span> <span m="4174670">property</span> <span m="4175840">that</span> <span m="4176760">some</span> <span m="4177300">digit</span> <span m="4178520">occurs</span> <span m="4178920">more</span> <span m="4179120">than</span> <span m="4179260">once.</span> <span m="4180540">So</span> <span m="4180670">it's</span> <span m="4180800">kind</span> <span m="4180970">of</span> <span m="4181050">interesting.</span> </p>
<p><span m="4181600">So</span> <span m="4182140">we</span> <span m="4182220">can</span> <span m="4182390">already</span> <span m="4182630">see</span> <span m="4182899">that</span> <span m="4183160">by</span> <span m="4183310">counting,</span> <span m="4185880">it's</span> <span m="4185960">sometimes</span> <span m="4186410">a</span> <span m="4186460">little</span> <span m="4186569">bit</span> <span m="4186729">counterintuitive.</span> <span m="4187569">Because</span> <span m="4188010">if</span> <span m="4188330">I</span> <span m="4188430">would</span> <span m="4189229">see</span> <span m="4189569">this</span> <span m="4189850">particular</span> <span m="4190550">property,</span> <span m="4191170">I</span> <span m="4191210">would</span> <span m="4191500">in</span> <span m="4191640">first</span> <span m="4191950">instance</span> <span m="4192479">think</span> <span m="4192800">that it's</span> <span m="4193080">a</span> <span m="4193270">very</span> <span m="4193740">special</span> <span m="4194240">property.</span> <span m="4194520">But</span> <span m="4194800">that's</span> <span m="4195010">not</span> <span m="4195170">true.</span> <span m="4195830">It's</span> <span m="4195900">very</span> <span m="4196110">common</span> <span m="4196605">it turns</span> <span m="4197100">out.</span> </p>
<p><span m="4200070">Now</span> <span m="4200260">a</span> <span m="4200320">special</span> <span m="4201610">case</span> <span m="4202220">of the</span> <span m="4202860">generalized</span> <span m="4203320">product</span> <span m="4203670">rule</span> <span m="4203860">is</span> <span m="4203960">the</span> <span m="4204060">product</span> <span m="4204460">rule.</span> <span m="4206120">And</span> <span m="4207510">this</span> <span m="4207730">is</span> <span m="4207890">defined</span> <span m="4208410">as</span> <span m="4208590">follows.</span> <span m="4210600">We</span> <span m="4210760">are going</span> <span m="4211100">to</span> <span m="4211150">first</span> <span m="4211400">of</span> <span m="4211500">all</span> <span m="4211740">define</span> <span m="4212210">a</span> <span m="4212300">product</span> <span m="4212690">over</span> <span m="4212990">sets.</span> <span m="4215940">The</span> <span m="4216110">definition</span> <span m="4216690">is</span> <span m="4217120">that</span> <span m="4218120">the</span> <span m="4218240">product</span> <span m="4218760">of</span> <span m="4219150">a</span> <span m="4219250">set</span> <span m="4219620">A1</span> <span m="4220670">with</span> <span m="4221170">A2</span> <span m="4222890">up to</span> <span m="4223750">An</span> <span m="4225550">is</span> <span m="4225750">actually</span> <span m="4226120">equal</span> <span m="4226540">to</span> <span m="4226650">the</span> <span m="4226760">set</span> <span m="4227020">of</span> <span m="4227170">sequences.</span> </p>
<p><span m="4233170">So</span> <span m="4233650">the</span> <span m="4233750">first</span> <span m="4234480">entry</span> <span m="4236740">is</span> <span m="4236930">selected</span> <span m="4237430">from</span> <span m="4237650">the</span> <span m="4237740">first</span> <span m="4238580">set,</span> <span m="4240930">the</span> <span m="4241030">second</span> <span m="4241440">entry</span> <span m="4242050">is</span> <span m="4242150">selected</span> <span m="4242590">from</span> <span m="4242760">the</span> <span m="4242820">second</span> <span m="4243260">set,</span> <span m="4244870">and</span> <span m="4245080">so</span> <span m="4245300">on.</span> <span m="4248630">And</span> <span m="4248930">now</span> <span m="4249090">the</span> <span m="4249210">product</span> <span m="4249600">rule</span> <span m="4249790">tells</span> <span m="4250180">us</span> <span m="4251550">by</span> <span m="4251680">just</span> <span m="4251950">applying</span> <span m="4252410">that</span> <span m="4252710">reasoning</span> <span m="4253240">over</span> <span m="4253560">here</span> <span m="4258470">that</span> <span m="4260820">the</span> <span m="4260950">cardinality</span> <span m="4262900">of</span> <span m="4263190">the</span> <span m="4263300">product</span> <span m="4263890">of</span> <span m="4264140">all those</span> <span m="4264500">sets</span> <span m="4267190">is</span> <span m="4267400">actually</span> <span m="4267780">equal</span> <span m="4269790">to</span> <span m="4271120">the</span> <span m="4271250">cardinality</span> <span m="4271800">of</span> <span m="4271890">the</span> <span m="4271970">first</span> <span m="4272350">set</span> <span m="4273740">multiplied</span> <span m="4274630">by</span> <span m="4274750">the</span> <span m="4274840">cardinality</span> <span m="4275460">of</span> <span m="4275580">the</span> <span m="4275640">second</span> <span m="4276030">set</span> <span m="4276570">and</span> <span m="4276770">so</span> <span m="4277020">on</span> <span m="4277440">up</span> <span m="4277860">to</span> <span m="4278280">the</span> <span m="4278390">cardinality</span> <span m="4278970">of</span> <span m="4279080">the</span> <span m="4279160">last</span> <span m="4279360">set.</span> <span m="4280210">Because</span> <span m="4280480">we</span> <span m="4280580">have</span> <span m="4282430">this</span> <span m="4282670">number</span> <span m="4282910">of</span> <span m="4283010">choices</span> <span m="4283760">for</span> <span m="4284010">the</span> <span m="4284140">very</span> <span m="4284400">first</span> <span m="4284760">element</span> <span m="4285120">over</span> <span m="4285370">here,</span> <span m="4285960">this</span> <span m="4286140">number</span> <span m="4286310">of</span> <span m="4286420">choices</span> <span m="4286840">for</span> <span m="4286950">the</span> <span m="4287040">second</span> <span m="4287430">one,</span> <span m="4287810">and</span> <span m="4287920">so</span> <span m="4288130">on.</span> </p>
<p><span m="4288730">And we</span> <span m="4288910">apply</span> <span m="4289250">that</span> <span m="4289430">rule,</span> <span m="4289650">and</span> <span m="4289760">we</span> <span m="4289880">see</span> <span m="4290100">that</span> <span m="4290250">this</span> <span m="4290450">is</span> <span m="4290580">the</span> <span m="4290770">result.</span> <span m="4293860">Now</span> <span m="4294090">when</span> <span m="4294210">we</span> <span m="4294280">use</span> <span m="4294700">this,</span> <span m="4295340">in</span> <span m="4295660">specific</span> <span m="4296210">to count</span> <span m="4296610">all</span> <span m="4296840">the</span> <span m="4297780">n-bit</span> <span m="4298150">sequences,</span> <span m="4299080">we</span> <span m="4299320">have</span> <span m="4300150">exactly</span> <span m="4301340">2</span> <span m="4301630">choices</span> <span m="4302040">for</span> <span m="4302160">the</span> <span m="4302240">first</span> <span m="4302810">bit.</span> <span m="4303970">We</span> <span m="4304070">have</span> <span m="4304240">to</span> <span m="4304340">set</span> <span m="4305010">0,1</span> <span m="4306330">in</span> <span m="4306510">our</span> <span m="4306730">example</span> <span m="4308160">times</span> <span m="4308670">the</span> <span m="4308770">set</span> <span m="4309040">0,</span> <span m="4309430">1</span> <span m="4309690">over</span> <span m="4310010">here</span> <span m="4311500">and</span> <span m="4311640">so</span> <span m="4311900">on.</span> </p>
<p><span m="4313520">And</span> <span m="4313690">that's</span> <span m="4314270">how</span> <span m="4314480">we</span> <span m="4314590">derived</span> <span m="4315410">that</span> <span m="4316880">we</span> <span m="4317070">have</span> <span m="4317520">2</span> <span m="4317780">times</span> <span m="4318150">2,</span> <span m="4318440">2</span> <span m="4318700">the</span> <span m="4318820">power</span> <span m="4320270">n</span> <span m="4320510">choices</span> <span m="4321270">for</span> <span m="4321620">an</span> <span m="4321800">n-bit</span> <span m="4322280">sequence.</span> <span m="4324670">So</span> <span m="4324730">now</span> <span m="4324880">we</span> <span m="4325010">come</span> <span m="4325250">to</span> <span m="4325620">the</span> <span m="4325920">sum</span> <span m="4326230">rule.</span> <span m="4327310">And</span> <span m="4327820">we will</span> <span m="4328100">give</span> <span m="4329160">an</span> <span m="4329270">example</span> <span m="4329730">for</span> <span m="4329940">that.</span> </p>
<p><span m="4331680">So</span> <span m="4331880">the</span> <span m="4331990">sum</span> <span m="4332240">rule</span> <span m="4332510">states</span> <span m="4333130">that</span> <span m="4333920">if</span> <span m="4334230">you</span> <span m="4334410">look</span> <span m="4334610">at</span> <span m="4334730">sets,</span> <span m="4335770">then</span> <span m="4335950">we</span> <span m="4336040">may</span> <span m="4336350">be</span> <span m="4336520">able</span> <span m="4336720">to</span> <span m="4336870">count their</span> <span m="4337240">union.</span> <span m="4338700">And</span> <span m="4338830">we</span> <span m="4338960">will</span> <span m="4339100">consider</span> <span m="4339470">a</span> <span m="4339630">very</span> <span m="4339700">specific</span> <span m="4340270">case.</span> <span m="4340610">In the next</span> <span m="4340930">lecture,</span> <span m="4341410">we</span> <span m="4341440">will</span> <span m="4341690">talk</span> <span m="4341990">about</span> <span m="4342240">the</span> <span m="4342320">general</span> <span m="4343320">case.</span> </p>
<p><span m="4345710">So</span> <span m="4345910">the</span> <span m="4346010">sum rule</span> <span m="4349780">is</span> <span m="4349980">the</span> <span m="4350050">following</span> <span m="4350930">counting</span> <span m="4352020">mechanism.</span> <span m="4353500">If</span> <span m="4354700">the</span> <span m="4354820">sets</span> <span m="4355410">A1</span> <span m="4357260">up</span> <span m="4357440">to</span> <span m="4357650">An</span> <span m="4359080">are</span> <span m="4359320">all</span> <span m="4360170">disjoint,</span> <span m="4361020">so</span> <span m="4361200">they</span> <span m="4361350">are</span> <span m="4361530">disjoint</span> <span m="4361870">sets,</span> <span m="4367600">then</span> <span m="4367890">we</span> <span m="4368010">know</span> <span m="4368290">that</span> <span m="4368450">if</span> <span m="4369740">I</span> <span m="4369890">try</span> <span m="4370220">to</span> <span m="4370380">count</span> <span m="4370690">the</span> <span m="4370920">union</span> <span m="4371350">of all</span> <span m="4371630">those</span> <span m="4371780">set,</span> <span m="4374580">it's</span> <span m="4374880">going</span> <span m="4375140">to</span> <span m="4375230">be</span> <span m="4375390">the sum</span> <span m="4375940">of</span> <span m="4376170">the</span> <span m="4376270">separate</span> <span m="4376690">cardinalities.</span> <span m="4379210">So</span> <span m="4380340">let</span> <span m="4380440">me</span> <span m="4380520">just</span> <span m="4380780">write</span> <span m="4380950">it</span> <span m="4381080">out</span> <span m="4381290">actually.</span> </p>
<p><span m="4382830">So</span> <span m="4383140">it's</span> <span m="4383740">the</span> <span m="4383860">cardinality</span> <span m="4384590">of</span> <span m="4385730">A1</span> <span m="4386780">plus</span> <span m="4387960">A2</span> <span m="4389760">all</span> <span m="4390000">the</span> <span m="4390100">way</span> <span m="4390760">to</span> <span m="4391280">An.</span> <span m="4392250">Why</span> <span m="4392550">is</span> <span m="4392680">this?</span> <span m="4393470">Well,</span> <span m="4394030">all the</span> <span m="4394460">sets</span> <span m="4394810">are</span> <span m="4394890">disjoint.</span> <span m="4396210">So</span> <span m="4397290">there</span> <span m="4397520">are</span> <span m="4397550">no</span> <span m="4397810">intersections</span> <span m="4398840">between</span> <span m="4399260">sets</span> <span m="4401270">that</span> <span m="4401460">contain</span> <span m="4401880">elements.</span> <span m="4402370">All the</span> <span m="4402560">intersections</span> <span m="4403240">are</span> <span m="4403380">empty.</span> </p>
<p><span m="4404580">So</span> <span m="4406160">counting</span> <span m="4406630">the</span> <span m="4406730">union</span> <span m="4407015">is</span> <span m="4407300">really</span> <span m="4407560">counting</span> <span m="4408000">each</span> <span m="4408280">separate</span> <span m="4408790">set.</span> <span m="4410310">And</span> <span m="4410530">that's</span> <span m="4410720">why</span> <span m="4410860">we</span> <span m="4410980">have</span> <span m="4411190">the</span> <span m="4411280">sum.</span> <span m="4412770">And</span> <span m="4412920">in the</span> <span m="4413000">next</span> <span m="4413320">lecture,</span> <span m="4413960">we</span> <span m="4413980">will</span> <span m="4414120">talk</span> <span m="4414370">about</span> <span m="4414740">inclusion,</span> <span m="4415070">exclusion</span> <span m="4415640">rule.</span> <span m="4416690">And</span> <span m="4416990">then</span> <span m="4417130">we will</span> <span m="4417350">take</span> <span m="4417580">into</span> <span m="4417830">account</span> <span m="4417900">that</span> <span m="4418280">we</span> <span m="4418660">have</span> <span m="4419070">intersections</span> <span m="4419920">that</span> <span m="4420060">are</span> <span m="4420140">not</span> <span m="4420390">empty.</span> </p>
<p><span m="4421020">But</span> <span m="4421130">let's</span> <span m="4421350">give</span> <span m="4421530">now</span> <span m="4421690">an</span> <span m="4421760">example</span> <span m="4423910">where</span> <span m="4424110">we</span> <span m="4428120">count</span> <span m="4428360">the</span> <span m="4428430">number</span> <span m="4428650">of</span> <span m="4428830">passwords</span> <span m="4431390">with</span> <span m="4431430">certain</span> <span m="4431570">properties.</span> <span m="4432900">And</span> <span m="4433730">we will</span> <span m="4433930">apply</span> <span m="4434380">all</span> <span m="4434570">these</span> <span m="4434720">different</span> <span m="4435050">rules</span> <span m="4435340">together.</span> <span m="4436900">And</span> <span m="4437030">that's</span> <span m="4437200">the</span> <span m="4437320">type</span> <span m="4437560">of</span> <span m="4437680">problems</span> <span m="4438220">that</span> <span m="4438340">you</span> <span m="4438450">would</span> <span m="4438570">like</span> <span m="4438800">to</span> <span m="4438910">be</span> <span m="4439020">able</span> <span m="4439200">to</span> <span m="4439310">solve.</span> </p>
<p><span m="4441350">So</span> <span m="4443980">in</span> <span m="4444100">our</span> <span m="4444260">last</span> <span m="4444540">example</span> <span m="4445010">here,</span> <span m="4445870">we</span> <span m="4446210">have</span> <span m="4446640">that</span> <span m="4446820">passwords</span> <span m="4448380">have</span> <span m="4448970">the</span> <span m="4449150">following</span> <span m="4450090">property.</span> <span m="4454850">They</span> <span m="4455210">are</span> <span m="4456280">6</span> <span m="4456590">to</span> <span m="4456730">8</span> <span m="4456950">symbols.</span> <span m="4462110">So</span> <span m="4462340">that's</span> <span m="4462560">property</span> <span m="4462980">1.</span> <span m="4465280">We</span> <span m="4465470">have</span> <span m="4465790">that</span> <span m="4465990">the</span> <span m="4466150">very</span> <span m="4466400">first</span> <span m="4466880">symbol</span> <span m="4467920">must</span> <span m="4468200">be</span> <span m="4469060">special</span> <span m="4469780">in</span> <span m="4469850">the</span> <span m="4469940">sense</span> <span m="4470400">that</span> <span m="4471580">it</span> <span m="4471740">is</span> <span m="4471920">a letter.</span> </p>
<p><span m="4477180">And</span> <span m="4477930">this</span> <span m="4478160">can</span> <span m="4478430">an</span> <span m="4478870">upper</span> <span m="4481270">or</span> <span m="4482140">lowercase.</span> <span m="4486830">And</span> <span m="4487400">say</span> <span m="4487690">that</span> <span m="4487960">the</span> <span m="4488170">other</span> <span m="4489870">symbols</span> <span m="4492250">are</span> <span m="4492580">actually</span> <span m="4494120">letters</span> <span m="4496290">or</span> <span m="4496510">digits.</span> <span m="4499180">So</span> <span m="4499370">let's</span> <span m="4499610">count</span> <span m="4499970">the</span> <span m="4500080">total</span> <span m="4500400">number</span> <span m="4500590">of</span> <span m="4500770">possible</span> <span m="4501390">passwords.</span> </p>
<p><span m="4503720">We're</span> <span m="4503840">going</span> <span m="4504030">to</span> <span m="4504150">use</span> <span m="4504350">the</span> <span m="4504410">sum</span> <span m="4504670">rule.</span> <span m="4506150">So</span> <span m="4506370">let's</span> <span m="4506750">define</span> <span m="4508680">what</span> <span m="4509020">kinds</span> <span m="4509350">of</span> <span m="4510130">sets</span> <span m="4510530">we</span> <span m="4510780">are</span> <span m="4511510">taking</span> <span m="4511890">the</span> <span m="4511970">symbols</span> <span m="4512420">from.</span> <span m="4513480">So</span> <span m="4513520">the</span> <span m="4513620">first</span> <span m="4513940">set</span> <span m="4514240">is</span> <span m="4514780">for</span> <span m="4515250">the</span> <span m="4515340">first</span> <span m="4515640">symbol,</span> <span m="4517220">which</span> <span m="4517410">we</span> <span m="4517620">call</span> <span m="4517910">f</span> <span m="4518770">or</span> <span m="4519140">first.</span> <span m="4521030">We</span> <span m="4521230">have</span> <span m="4521770">all</span> <span m="4521980">the</span> <span m="4522060">letters</span> <span m="4522510">a,</span> <span m="4522925">b, c</span> <span m="4523340">in</span> <span m="4523430">lowercase,</span> <span m="4525200">and</span> <span m="4525530">then</span> <span m="4525920">all</span> <span m="4526130">of</span> <span m="4526260">them</span> <span m="4526600">in</span> <span m="4526790">uppercase.</span> <span m="4530430">And</span> <span m="4530650">in</span> <span m="4530750">total,</span> <span m="4532700">we</span> <span m="4532870">have</span> <span m="4533310">52</span> <span m="4534710">elements</span> <span m="4535240">in</span> <span m="4535350">this</span> <span m="4535560">set.</span> </p>
<p><span m="4537900">For</span> <span m="4538060">the</span> <span m="4538150">second</span> <span m="4538510">symbol,</span> <span m="4540130">or</span> <span m="4540220">the</span> <span m="4540320">other</span> <span m="4540500">symbols,</span> <span m="4541600">we</span> <span m="4541730">have</span> <span m="4542410">all</span> <span m="4542800">these</span> <span m="4543830">letters,</span> <span m="4545010">but</span> <span m="4545150">also</span> <span m="4545760">all</span> <span m="4545900">the</span> <span m="4545990">digits,</span> <span m="4546870">0,</span> <span m="4547290">1,</span> <span m="4548570">up</span> <span m="4548760">to</span> <span m="4548900">9.</span> <span m="4550020">And</span> <span m="4552000">this</span> <span m="4552140">set</span> <span m="4552780">actually</span> <span m="4553290">has</span> <span m="4553840">cardinality</span> <span m="4554450">62.</span> <span m="4555260">We</span> <span m="4555440">added</span> <span m="4556100">10</span> <span m="4556460">digits.</span> </p>
<p><span m="4558960">So</span> <span m="4559170">let's</span> <span m="4559430">talk</span> <span m="4559710">about--</span> <span m="4564710">actually,</span> <span m="4565130">we</span> <span m="4565260">like</span> <span m="4565450">to</span> <span m="4565560">use</span> <span m="4565860">this</span> <span m="4567560">in</span> <span m="4567720">the</span> <span m="4567900">sum rule</span> <span m="4568386">as well.</span> <span m="4578600">So</span> <span m="4578840">how</span> <span m="4578960">do</span> <span m="4579040">we</span> <span m="4579190">count?</span> <span m="4580270">What</span> <span m="4580410">kind</span> <span m="4580560">of</span> <span m="4580620">possibilities</span> <span m="4581220">do</span> <span m="4581400">we</span> <span m="4581550">really</span> <span m="4581800">have?</span> <span m="4582150">So</span> <span m="4582330">let's</span> <span m="4582740">describe</span> <span m="4583130">the</span> <span m="4583200">set</span> <span m="4583690">of</span> <span m="4584430">passwords</span> <span m="4585170">explicitly</span> <span m="4586020">in</span> <span m="4586150">a</span> <span m="4586200">formula.</span> <span m="4587940">So</span> <span m="4588170">let</span> <span m="4588690">p</span> <span m="4589780">be</span> <span m="4589950">the</span> <span m="4590030">set</span> <span m="4590310">of</span> <span m="4591420">possible</span> <span m="4591850">passwords.</span> </p>
<p><span m="4598110">And</span> <span m="4599590">this</span> <span m="4599840">one</span> <span m="4600030">is</span> <span m="4600160">actually</span> <span m="4600430">equal</span> <span m="4600920">to--</span> <span m="4601920">well,</span> <span m="4602390">I</span> <span m="4602520">need</span> <span m="4602760">to</span> <span m="4602870">choose</span> <span m="4603170">a</span> <span m="4603280">first</span> <span m="4603670">symbol.</span> <span m="4605050">And</span> <span m="4605270">then</span> <span m="4606050">I</span> <span m="4606200">need</span> <span m="4606470">to</span> <span m="4606700">choose</span> <span m="4607050">a</span> <span m="4607110">second</span> <span m="4607530">symbol,</span> <span m="4609100">and</span> <span m="4609290">a</span> <span m="4609370">third,</span> <span m="4611070">and</span> <span m="4611250">a</span> <span m="4611300">fourth,</span> <span m="4612520">and</span> <span m="4613060">a</span> <span m="4613170">fifth,</span> <span m="4614560">and a</span> <span m="4614870">sixth.</span> <span m="4615600">That's</span> <span m="4615800">one</span> <span m="4615990">possibility.</span> <span m="4616690">I</span> <span m="4616800">use</span> <span m="4617040">6</span> <span m="4617390">symbols.</span> </p>
<p><span m="4618780">I</span> <span m="4618850">can</span> <span m="4619020">also</span> <span m="4619190">use</span> <span m="4619430">7</span> <span m="4619800">or</span> <span m="4619910">8</span> <span m="4620140">symbols.</span> <span m="4620660">But</span> <span m="4620800">this</span> <span m="4620990">is</span> <span m="4621110">one</span> <span m="4621340">of</span> <span m="4621460">them.</span> <span m="4624770">We</span> <span m="4624810">also</span> <span m="4625120">denote</span> <span m="4625500">this</span> <span m="4626730">as</span> <span m="4627310">S</span> <span m="4628390">to</span> <span m="4628790">the</span> <span m="4628910">power</span> <span m="4629180">5.</span> <span m="4630120">That's</span> <span m="4630380">an</span> <span m="4630490">equivalent</span> <span m="4630990">notation.</span> </p>
<p><span m="4635110">The</span> <span m="4635240">other</span> <span m="4635480">possibilities</span> <span m="4636500">for</span> <span m="4637420">passwords</span> <span m="4637840">are</span> <span m="4638230">that we</span> <span m="4638620">first</span> <span m="4638980">choose</span> <span m="4640940">an</span> <span m="4641200">entry</span> <span m="4641560">from</span> <span m="4641800">f,</span> <span m="4642610">a</span> <span m="4643010">letter.</span> <span m="4644510">And</span> <span m="4644760">then</span> <span m="4645580">we</span> <span m="4645740">will</span> <span m="4645880">need</span> <span m="4646090">to</span> <span m="4646210">choose</span> <span m="4646540">another</span> <span m="4646810">6</span> <span m="4647160">symbols.</span> <span m="4648460">In</span> <span m="4648640">total,</span> <span m="4649270">we</span> <span m="4649440">have</span> <span m="4649650">7.</span> </p>
<p><span m="4651940">And</span> <span m="4652130">we</span> <span m="4652260">have</span> <span m="4652370">another</span> <span m="4652650">possibility</span> <span m="4653320">where</span> <span m="4653430">we</span> <span m="4653780">choose</span> <span m="4654100">a</span> <span m="4654170">first</span> <span m="4654500">symbol,</span> <span m="4655790">and</span> <span m="4656030">then</span> <span m="4656210">we</span> <span m="4656330">choose</span> <span m="4657300">7</span> <span m="4657680">other</span> <span m="4657870">symbols.</span> <span m="4659290">So</span> <span m="4659530">has</span> <span m="4659770">8</span> <span m="4660130">symbols.</span> <span m="4660570">This has</span> <span m="4661010">in total</span> <span m="4661390">7.</span> <span m="4661880">This</span> <span m="4662060">one</span> <span m="4662240">has</span> <span m="4662440">in total</span> <span m="4662830">6</span> <span m="4663130">symbols.</span> </p>
<p><span m="4663970">So</span> <span m="4664620">this</span> <span m="4664860">covers</span> <span m="4665210">all</span> <span m="4665630">the</span> <span m="4665760">possible</span> <span m="4666190">passwords.</span> <span m="4667730">So</span> <span m="4667920">let's</span> <span m="4668140">count</span> <span m="4668440">them.</span> <span m="4669410">We</span> <span m="4669530">know that</span> <span m="4670220">these</span> <span m="4670520">sets</span> <span m="4671120">are</span> <span m="4671380">all</span> <span m="4671670">the</span> <span m="4671750">different.</span> <span m="4672730">They</span> <span m="4672890">are</span> <span m="4672980">distinct.</span> </p>
<p><span m="4674590">These</span> <span m="4674810">are</span> <span m="4674910">sequences</span> <span m="4675580">that</span> <span m="4675730">have</span> <span m="4676380">6</span> <span m="4676750">entries,</span> <span m="4677850">7,</span> <span m="4678390">and</span> <span m="4678580">8</span> <span m="4678810">entries.</span> <span m="4679200">So</span> <span m="4679840">if</span> <span m="4680040">you</span> <span m="4680170">look</span> <span m="4680340">at</span> <span m="4680400">the</span> <span m="4680490">cardinality</span> <span m="4682420">of</span> <span m="4682755">P,</span> <span m="4684340">it's</span> <span m="4684550">actually</span> <span m="4684870">equal</span> <span m="4685870">to</span> <span m="4685990">the</span> <span m="4686140">sum</span> <span m="4686660">by</span> <span m="4686820">the</span> <span m="4686920">sum</span> <span m="4687170">rule</span> <span m="4687710">that</span> <span m="4687840">we</span> <span m="4688095">just</span> <span m="4688540">described</span> <span m="4689160">over</span> <span m="4689380">there,</span> <span m="4692130">is</span> <span m="4692370">equal</span> <span m="4692730">to</span> <span m="4694360">the</span> <span m="4694530">very</span> <span m="4694800">first</span> <span m="4695260">one,</span> <span m="4696050">f</span> <span m="4696610">times</span> <span m="4697140">S</span> <span m="4697380">to</span> <span m="4697480">the</span> <span m="4697610">power</span> <span m="4697900">5</span> <span m="4698490">plus</span> <span m="4699430">the</span> <span m="4699550">cardinality</span> <span m="4700320">of</span> <span m="4702850">the</span> <span m="4703000">product</span> <span m="4703320">of</span> <span m="4703450">f</span> <span m="4703760">with</span> <span m="4704040">S</span> <span m="4704430">to</span> <span m="4704550">the</span> <span m="4704670">power</span> <span m="4705020">6.</span> <span m="4706800">And</span> <span m="4707570">we</span> <span m="4707800">have</span> <span m="4709550">f</span> <span m="4713940">times</span> <span m="4714990">S</span> <span m="4715170">to</span> <span m="4715260">the</span> <span m="4715360">power</span> <span m="4715690">7.</span> </p>
<p><span m="4717180">So</span> <span m="4717320">this</span> <span m="4717540">is</span> <span m="4717630">simply</span> <span m="4718050">by</span> <span m="4718190">application</span> <span m="4718970">of</span> <span m="4719240">the</span> <span m="4720160">sum</span> <span m="4720470">rule.</span> <span m="4721580">And</span> <span m="4721770">now</span> <span m="4721910">we</span> <span m="4722030">can</span> <span m="4722220">apply</span> <span m="4722570">the</span> <span m="4722680">product</span> <span m="4723080">rule</span> <span m="4723390">very</span> <span m="4723680">simply,</span> <span m="4724630">which</span> <span m="4724820">is</span> <span m="4724940">this</span> <span m="4725150">one.</span> <span m="4726060">So</span> <span m="4726300">that's</span> <span m="4727690">equal</span> <span m="4728090">to</span> <span m="4728180">the</span> <span m="4728310">cardinality</span> <span m="4729120">of</span> <span m="4729320">f</span> <span m="4732328">times</span> <span m="4732820">the</span> <span m="4732910">cardinality</span> <span m="4733620">of</span> <span m="4733850">S</span> <span m="4734250">to</span> <span m="4734360">the</span> <span m="4734480">power</span> <span m="4734740">5.</span> <span m="4736100">And</span> <span m="4736250">then</span> <span m="4736370">we</span> <span m="4736480">have</span> <span m="4736710">the</span> <span m="4736800">same</span> <span m="4737160">rule</span> <span m="4737680">applied</span> <span m="4738330">to</span> <span m="4738560">this</span> <span m="4738840">one.</span> </p>
<p><span m="4739550">It's</span> <span m="4739630">the</span> <span m="4739730">commonality</span> <span m="4740410">of</span> <span m="4740590">f</span> <span m="4740980">times</span> <span m="4741290">the</span> <span m="4741360">cardinality</span> <span m="4741980">of</span> <span m="4742130">S</span> <span m="4742750">to</span> <span m="4743000">the</span> <span m="4743120">power</span> <span m="4743420">6.</span> <span m="4744840">And</span> <span m="4745000">then</span> <span m="4745150">we have</span> <span m="4745340">the</span> <span m="4745420">cardinality</span> <span m="4746060">of</span> <span m="4746210">f</span> <span m="4747090">times</span> <span m="4747500">the cardinality</span> <span m="4748180">of</span> <span m="4748340">S</span> <span m="4748590">to</span> <span m="4748700">the</span> <span m="4748810">power</span> <span m="4749080">7.</span> <span m="4752190">Now,</span> <span m="4752370">we</span> <span m="4752530">simply</span> <span m="4753000">plug-in</span> <span m="4753420">these</span> <span m="4753630">numbers.</span> </p>
<p><span m="4754820">And</span> <span m="4755110">then you</span> <span m="4755200">will have</span> <span m="4755490">the</span> <span m="4756180">total</span> <span m="4756530">number</span> <span m="4756820">of</span> <span m="4756960">passwords</span> <span m="4759090">that</span> <span m="4759210">you</span> <span m="4759330">can</span> <span m="4759480">select</span> <span m="4759830">from.</span> <span m="4760020">It</span> <span m="4760110">turns</span> <span m="4760440">out</span> <span m="4760690">to</span> <span m="4760890">be</span> <span m="4761870">about</span> <span m="4762410">1.8</span> <span m="4763960">times</span> <span m="4764610">10</span> <span m="4764780">to</span> <span m="4764900">the</span> <span m="4765030">power</span> <span m="4765350">14.</span> <span m="4767640">So</span> <span m="4767750">here</span> <span m="4767910">we</span> <span m="4768020">have</span> <span m="4768150">applied</span> <span m="4768470">both</span> <span m="4768730">to</span> <span m="4768830">sum rule</span> <span m="4769250">and</span> <span m="4769340">the</span> <span m="4769440">product</span> <span m="4769770">rule.</span> </p>
<p><span m="4770320">So</span> <span m="4770490">in</span> <span m="4770560">general,</span> <span m="4770980">you</span> <span m="4771110">will</span> <span m="4771230">see</span> <span m="4771830">that</span> <span m="4771930">you have to</span> <span m="4772220">apply</span> <span m="4772510">multiple</span> <span m="4773020">rules</span> <span m="4773290">together</span> <span m="4773620">in</span> <span m="4773860">order</span> <span m="4774140">to</span> <span m="4775540">find an</span> <span m="4775920">answer</span> <span m="4776215">to your</span> <span m="4776510">counting</span> <span m="4776900">problem.</span> <span m="4777750">And</span> <span m="4777880">you'll</span> <span m="4778020">see</span> <span m="4778170">that</span> <span m="4778300">on</span> <span m="4778410">a</span> <span m="4778460">problem</span> <span m="4778800">set.</span> <span m="4779600">And</span> <span m="4779730">we will</span> <span m="4779900">give</span> <span m="4780100">a</span> <span m="4780150">few</span> <span m="4780360">more</span> <span m="4780540">examples</span> <span m="4781210">in</span> <span m="4781370">next</span> <span m="4781640">lecture.</span> </p>
<p><span m="4782510">And</span> <span m="4782640">we</span> <span m="4783500">will</span> <span m="4783640">start</span> <span m="4783990">talking</span> <span m="4784390">about</span> <span m="4784650">the</span> <span m="4784730">generalization</span> <span m="4785550">of</span> <span m="4785670">the</span> <span m="4785750">sum</span> <span m="4786000">room</span> <span m="4786330">called</span> <span m="4786680">inclusion</span> <span m="4787120">exclusion.</span> <span m="4788170">And</span> <span m="4788340">we will</span> <span m="4788610">give</span> <span m="4788770">you</span> <span m="4788920">another</span> <span m="4789280">type</span> <span m="4789560">of</span> <span m="4789700">proof</span> <span m="4789970">technique</span> <span m="4790750">called</span> <span m="4792220">combinatorial</span> <span m="4793190">proofs.</span> <span m="4794330">All</span> <span m="4794470">right.</span> <span m="4794530">Good</span> <span m="4794780">luck</span> <span m="4794980">with</span> <span m="4795060">the</span> <span m="4795140">problems.</span> </p>
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